A note on the counterfeit coins problem

Transcription

1 A note on the counterfeit coins problem Li An-Ping Beijing , P.R.China Abstract In this paper, we will present an algorithm to resolve the counterfeit coins problem in the case that the number of false coins is unknown in advance. Keywords: counterfeit coins problem, combinatorial search, information theoretic bound.

2 The counterfeit coins problem is a well-known combinatorial search problem, namely, search for the counterfeit coins in a number of coins by a balance, where the fakes and the normals have same semblance but different weights. The problem has several versions, for instance, the fakes are known lighter or heavier than the normals, and the number of the fakes is assumed known. For the detail results and researches in this subject refer to see the papers [1]~[7]. In this paper, we will consider a more general case that the number of the fakes is unknown beforehand. In general, it is assumed that the coins will be permitted to be remarked by numbers in order to be distinguished each other. Our main result is as following Proposition 1. Suppose that S is a set of n coins with same semblance, in which possibly there are some counterfeit coins, which are heavier (or lighter) than the normals. Denoted by gnthe ( ) least number of weighings need to sort S into the normals and fakes by a balance, assumed that additional normal coins will be available if needed, then it has but with a exception that g (3) = 3. n log 2 g( n) 7 n/11, 3 (1) The proof of Proposition 1 will largely rely on the following lemma, which is one of our unpublished results obtained in Lemma 1. Let A be a set of 11 coins with two possible distinct weights, then A can be sorted by a balance with at most 7 weighings, and in the sixth weighing it will be found if A contains only one kind weight of coins. Proof. It will be suffice to provide a feasible algorithm for prove the lemma. We have succeeded such an algorithm, which sketch is provided as an appendix in the end of the paper, for to describe the whole algorithm require much more space. The readers are encouraged yet to design own algorithms for it as a puzzle. Lemma 2. Let S be same as in Proposition 1, S= A B, B = 3. If there is a algorithm F with at most k weighings, k 1, to find all the fakes in A, then there is also a algorithm F to find all the fakes in S with at most k + 2 weighings. Proof. We apply the algorithm F on set A up to k 1 weighings. Let Γ be a non-empty output after k 1 weighings, clearly, Γ = 1,2,or 3. If Γ= 1, that is, Γ contains only one objective, and so which is the set of all the fakes in A, the rest we need to do is to find the fakes in set B, thus in this case, g( S ) k 1 + g(3) k+ 2.

3 If Γ= 2, suppose that Γ= { X, Y}, then there is one coin x x X(or Y) such that x Y(or X), let B = x B, it is easy to know we are provided to search for all the fakes in set B. On the other hand, it is not difficult to know that g (4) = 3, hence, g( S ) k 1 + g(4) k+ 2. If Γ= 3, suppose that Γ= { X, Y, Z}. If there are two of { X, Y, Z }, say, X and Y, such that X / Y Z and Y / X Z, then take x X \( Y Z) and y Y \( X Z). It is easy to know that which one of three objectives in the set Δ = {, xy, } is fakes will tell which of { X, YZ, } is the set of fakes in A. So it will be suffice to search all the fakes in sets Δ and B in three weighings, such an algorithm is described in Page 21. Thereby, assume that X Y Z and Y X Z. If there is no one of Z X and Z Y is contained in the other one, then take x ( Z X)\( Z Y), y ( Z Y)\( Z X). Similar to the above, which one of three objectives in the set Δ = {, x yx, y} is fakes will tell which of { X, Y, Z} is the set of fakes in A. We have to find all the fakes in sets Δ and B in three weighings. The algorithm is presented in Page 22. If ( Z X) ( Z Y), this implies that X Y. Moreover, from Y = X ( Z Y), it has ( Z Y) X. If X / ( Z Y), then take x X \( Z Y) and z ( Z Y)\ X, then Δ = {,, x zxz }, obviously, the case is similar to Δ. So, the rest to be discussed is X ( Z Y). From Y X Z, it has Y Z. Take y Y \ X and z Z \ Y, then which one of three objectives in the set Δ = { z y, y, } is fakes will tell which of { X, YZ, } is the set of fakes in A. We have to find all the fakes in sets Δ and B in three weighings. The sketch of such an algorithm has been shown in the page 23 of the appendices. Proof of Proposition 1. The left-hand of (1) is trivial for the information theoretical bound of gn ( ) is equal to n log3 2, so we are provided to prove the right-hand of (1). It is not difficult to verified that for n 11, 3 n, it has gn ( ) 7 n/11, and g (3) = 3. For

4 n > 11, let n= 11 m+ r, 0 r < 11, and r 3, by Lemma 1, there is S= A i, Ai = 11, 0 i< m, A m = r. If 0 i m gn ( ) 7m+ 7 r/11 7 n/11. If r = 3, by Lemma 1 and Lemma 2, it has gn ( ) 7m+ 2 7m+ 7 3/11 7 n/11. In the following, we will investigate the sorting case, that is, without provided of additional normal coins. Proposition 2. Suppose that S is a set of n coins with same semblance and with two possible distinct weights. Denoted by gn ( ) the least number of weighings needed to sort S in the weights by a balance, then it has log (2 1) gn ( ) 7 n/11. n 3 (2) To prove Proposition 2, it needs following initial result Lemma 3. For n 11, there is an algorithm with at most 7 n /11 and it will be found if all the coins in S are of same weight in the ( 7 n /11 1) except n = 3. weighings to sort S, -th weighing Proof. It is easy to verify the conclusions in the Lemma for n < 6 directly. Moreover, the required algorithm for n = 7 may be derived from the one for n = 6, and the algorithm for n = 8 may be derived from the one for n = 7. The required algorithm for n = 9 may be obtained from the algorithms for n = 6 and Lemma 2, the one for n = 10 may be derived from the one for n = 9. The case n = 11 is as in Lemma 1. So, the rest algorithm to be designed is the one for n = 6, such an algorithm is provided in the page 24. The Proof of Proposition 2. The lower bound of (2) is the information theoretic bound of gn ( ), so, to be proved is the upper bound of (2). With Lemma 3, we may assume that n > 11. Let n= 11 k+ r, 0 r < 11, and Denoted by 1 i k S = A B, A = 11,1 i k, B = r. 1 i k i i Q = A i, then by Lemma 1, we can spend at most 7k weighings to complete a sorting of the set Q, and if r 3, by Lemma 3, at most 7k+ 7 r/11 weighings will be

5 enough to complete a sorting of whole set S, hence gn ( ) 7k+ 7 r/11 = 7 n/11. So, the rest to be discussed is the case r = 3. With Lemma 2, it spends at most 7k + 2 weighings to complete a sorting of the set S, hence So, the proof is finished. gn ( ) 7k+ 2 = 7k+ 7 3/11 = 7 n/11. It maybe should be mentioned that in Lemma 2 the third algorithm we presented, that is the last one to deal with the case X Y Z, will need a normal coin, clearly which may be provided unless Z =Q, however, this case will not occur in applying the algorithm described in Lemma 1, for we will find whether the coins in set Q are of same weight by 7k 1( = 6k+ k 1) weighings if it is, so, the objective set Γ = { Q }, rather than Γ = { X, Y, Z}. In addition, as a derivative result, there is a corollary of Lemma 1 Corollary 1. Let A i,1 i 11, be 11 sets of coins, each Ai consists of two coins, one is normal and one is fake, the normals and the fakes have distinct weights, then at most spend 7 weighings by a balance to find all the fakes in A i s, 1 i 11. Proof. From each A take a coin,1 i 11, to form a coin set A, applying Lemma 1 to A, i if there are two kinds of coins in A, then each A,1 i 11, will be sorted, otherwise, that is, i if A contains only one kind of coins, then a additional weighing will work well. Likely, there may be the algorithms independent of Lemma 1 for the result of Corollary 1, which will be left to the readers. Acknowledgement: The author greatly appreciates Professor L. Lovasz for his endorsement for this paper, and many thanks to arxiv for publish it. Finally, the author should acknowledge Hagen von Eitzen for he had pointed out the flaws in the statement and the proof of Proposition 1 in the earlier versions, and told me the counter-example n = 19 for the conjecture gn ( ) = n log3 2 once proposed in the original version of this paper.

6 References [1] M. Aigner, Combinatorial Search. Wiley-Teubner 1988 [2] R. Bellman, B. Glass, On various versions of the defective coin problem. Inform. Control 4, (1961) [3] Richard K. Guy; Richard J. Nowakowski, Coin-Weighing Problems, The American Mathematical Monthly, Vol. 102, No. 2. (Feb., 1995), pp [4] A.P. Li, On the conjecture at two counterfeit coins, Disc. Math. 133 (1994), [5] A.P. Li, Three counterfeit coins problem, J. Comb. Theory Ser. A, 66 (1994), [6] A.P. Li, M. Aigner, Searching for counterfeit coins, Graphs Comb. 13 (1997), 9-20 [7] B. Manvel, Counterfeit coin problem, Math. Mag. 50, (1977) [8] L. Pyber, How to find many counterfeit coins? Graphs Comb. 2, (1986)

7 Appendices Algorithm to Sort 11 Coins The expression L : R means a comparison or a weighing with L and R are the left-hand and the right-hand of the balance respectively, and for simplicity, the expression {1,2,3} means the total weight of the coins with marks 1,2 and 3, and so on. Fig.22 (<) Fig.21 (=) {3,4,7,10 } :{1,2,6,11} (<) {5,7,9 } :{6,8,10 } (<) {1,7,8 } :{2,9,10} (<) {1,2,3 } :{4,5,6 } (>) (=) Fig.20 {2,9,11 } :{ 3,4,5} (=) (<) (=) (>) Fig.19 Fig.18 Fig.17 Fig.16 (<) Fig.15 (=) {2,3,6} :{5,8,9} (<) {1,5,7,9}:{2,6,8,10} (=) (>) (=) Fig.14 {1,2,4} :{ 3,7,10} (<) (=) (>) Fig.13 Fig.12 Fig.11 Fig.10 (<) Fig.9 (=) {1,5,6}:{2,3,7} (<) {7,9}:{8,10} (<) {1,7.8}:{ 4,9,10} (>) Fig.8 (=) Fig.7 (<) {5,7,10}:{ 6,8,9} (=) (=) Fig..6 Fig.5 (<) Fig.4 (=) {1,3,4}:{8,9,11} (<) {1,2,7}:{ 3,4,8} (>) Fig.3 (=) Fig.2 (<) {1,3 } :{ 2,4} (=) Fig.1 Fig. 0

8 ( 7,8,9,10) ( 1,4) (>) (>) {1 } :{2 } (=) {10 } :{ 11} (<) {1,2 } :{10,11 } (>) {1 } :{ 2} (=) {10 } :{11 } (<) (=) (<) ( 11 ) ( 2,3,5,6) {1 } :{ 2} (<) (=) (>) {10 } :{11 } (~) {10 } :{11 } Fig.1 {6}:{11} {10}:{ 11} (>) (>) {9 }:{10 } (=) {3,5 } :{1,11 } (<) {1 } :{7 } (>) {5,11 } :{6,9 } (=) {5 } :{ 6} (<) (=) (<) {9}:{10} {10}:{11} { 5,11} :{6,9} (<) (=) (>) {10}:{11} {9}:{10} {10}:{11} Fig.2 {5,9}:{7,10} { 6,9} :{ 7,10} (>) (>) { 10} :{11} (=) {1,9}:{8,11} (<) {5}:{6} (>) {1,9}:{8,11} (=) {10}:{11} (<) (=) (<) {1}:{ 7} {1}:{7} {2} :{ 7} (<) (=) (>) { 10} :{11} {1}:{3} { 9} :{11} Fig.3

9 { 2} :{ 3} { 2} :{3} (>) (>) {1}:{2} (=) {9}:{ 10} (<) {5 } :{6 } (>) {9 } :{10 } (=) {1 } :{ 2} (<) (=) (<) {2}:{3} { 2} :{ 3} {9 } :{ 11} (<) (=) (>) {3}:{ 10} {9}:{10} {3}:{ 4} Fig.4 {9}:{11} {7}:{ 8} (>) (>) {10}:{11} (=) {3,6}:{7,9} (<) {3,5}:{6,9} (>) {9}:{ 11} (=) { 2} :{10} (<) (=) (<) {1,4}:{10,11} { 2} :{3} {9 } :{10} (<) (=) (>) {2,10}:{3,11} {1}:{2} {2}:{11} Fig.5 {3}:{11} {2}:{11} (>) (>) {1}:{7} (=) {3,11}:{9,10} (<) { 2} :{3} (>) {2,11}:{9,10} (=) {1}:{7} (<) (=) (<) {1}:{7} {1}:{7} {1,11 } :{2,3 } (<) (=) (>) {1}:{11} {6}:{ 9} {1}:{11} Fig.6

10 {1}:{11} {1,4}:{10,11} (>) (>) {6}:{8} (=) {1,11}:{2,8} (<) {2,5}:{3,7} (>) {1,6}:{5,11} (=) { 6} :{8} (<) (=) (<) {4,5}:{10,11} {5}:{11} {1,10} :{8,11} (<) (=) (>) {8}:{11} {6}:{ 9} {2}:{11} Fig.7 {2,3}:{9, 11} {2,3}:{9,11} (>) (>) {9 } :{11} (=) {5,9 } :{6,11} (<) {2} :{3} (>) {5,9 } :{6,11} (=) {9 } :{11} (<) (=) (<) {2,3}:{9,11} {2,3}:{9,11} {5,6} :{9,11} (<) (=) (>) {9}:{ 11} {9}:{10} {9}:{11} Fig.8 {2,3}:{9,11} {2,3}:{9,11} (>) (>) {9 } :{11} (=) {5,9} :{6,11} (<) {2} :{3} (>) {5,9 } :{6,11} (=) {9 } :{11} (<) (=) (<) {2,3}:{9,11} {2,3}:{9,11} {5,6} :{9,11} (<) (=) (>) {9}:{ 11} {9}:{10} {9}:{11} Fig.9

11 {10}:{11} {10}:{11} (>) (>) {5}:{6} (=) {1,3,11}:{ 2,5,6} (<) {1,10} :{8,9} (>) {1,3,11} :{ 2,5,6} (=) {5 } :{6 } (<) (=) (<) {5}:{6} {5}:{6} {1,3,11} :{ 2,5,6} (<) (=) (>) {5}:{6} {5}:{6} {10}:{11} Fig.10 {1,11}:{3,4} {2}:{11} (>) (>) {2}:{11} (=) { 1,5} :{7,11} (<) {1,2}:{8,9} (>) { 1,10} :{2,11} (=) { 2} :{ 11} (<) (=) (<) {2}:{4} {7}:{11} {1,2} :{10,11 } (<) (=) (>) { 4} :{5} {3,4}:{ 6,7} {10}:{11} Fig.11 {8}:{11} {8}:{ 11} (>) (>) { 6} :{8} (=) {3,4}:{ 9,11} (<) {1}:{2} (>) {3,4}:{9,11} (=) {10}:{11} (<) (=) (<) {8}:{11} { 8} :{11} {3,4} :{ 9,11} (<) (=) (>) {8}:{11} {6}:{8} {8}:{11} Fig.12

12 {3,4}:{9,11} {9}:{11} (>) (>) {3}:{4} (=) {3,8}:{4,11} (<) {3,7,10}:{4,5,6} (>) {3,5}:{9,11} (=) { 3} :{4} (<) (=) (<) {3,4}:{9,11} {9}:{11} {3,4} :{9,11} (<) (=) (>) {9}:{11} { 3} :{5} {9}:{11} Fig.13 {7}:{11} {10}:{11} (>) (>) {2,4}:{5,6} (=) {2}:{10} (<) {3,5,10}:{7,9,11} (>) {2,8}:{ 10,11} (=) {1,3}:{7,8} (<) (=) (<) {4}:{6} { 7} :{8} {2,4} :{10,11 } (<) (=) (>) {1}:{3} {10}:{ 11} {3,4}:{7,8} Fig.14 {3}:{11} {1,11}:{4, 7} (>) (>) {10}:{ 11} (=) {5,11}:{7,9} (<) {2,5,9}:{3,4,7} (>) {1,4}:{7,11} (=) { 2} :{ 4} (<) (=) (<) {5,9}:{7,11} {7}:{11} {2,9} :{3,11 } (<) (=) (>) {3}:{11} { 10} :{11} {10}:{ 11} Fig.15

13 {3,4}:{5,6} {4}:{5} (>) (>) {2}:{ 3} (=) {1,5}:{ 7,11} (<) {1,6,11}:{3,7,10} (>) {5,10}:{2,11} (=) {1,4}:{5,6} (<) (=) (<) {4}:{ 6} {4}:{6} {5} :{ 6} (<) (=) (>) {4}:{6} {4}:{ 6} {4}:{ 11} Fig.16 {8}:{ 10} {1}:{8} (>) (>) {2}:{7} (=) {2,9}:{ 10,11} (<) {1,4,11}:{3,5,10} (>) {5,6}:{7,11} (=) {2,4}:{5,6} (<) (=) (<) {7}:{ 8} {2,6}:{4,5} {7,9} :{8,11} (<) (=) (>) {6}:{8} { 1} :{ 7} {2}:{11} Fig.17 {5}:{ 7} {1}:{7} (>) (>) {3}:{4} (=) {1,6}:{5,8} (<) {2,6,8}:{3,5,11} (>) {1,6}:{ 8,9} (=) {2}:{7} (<) (=) (<) {2}:{8} {6}:{10} {4} :{ 9} (<) (=) (>) {1}:{8} {10}:{11} {2}:{ 7} Fig.18

14 {3,11}:{4,10} { 1,2} :{3,11} (>) (>) {2}:{11} (=) {3,6}:{7,10} (<) {1,5,9} :{2,4,11} (>) {1,4} :{8,9 } (=) {2,5 } :{3,4 } (<) (=) (<) {6}:{ 11} { 3} :{ 4} {1,4} :{2,6 } (<) (=) (>) {8}:{10} {5}:{7} {2,4}:{3,8} Fig.19 {2}:{ 11} {2,3}:{4,8} (>) (>) {2}:{3} (=) { 8,9} :{3,11} (<) {1,5,6} :{4,7,11} (>) {2,4} :{3,9 } (=) {3 } :{9 } (<) (=) (<) {3,6}:{4,9} {3,4}:{8,9} {8,9} :{6,11} (<) (=) (>) {2,4}:{3,5} {2,4}:{ 3,5} {2}:{3} Fig.20 {5,6}:{7,8} { 8} :{10} (>) (>) {5}:{8} (=) {2,4}:{5,9} (<) {1,2} :{7,11} (>) {5} :{9 } (=) {6,7} :{8,9 } (<) (=) (<) {3}:{8} {1}:{ 8} {2,8} :{7,10 } (<) (=) (>) { 5} :{9} {5}:{9} {3,6}:{5,9} Fig.21

15 {6,7}:{ 8,9} {6,7}:{9,11} (>) (>) {6,7}:{8,9} (=) { 1,2} :{ 4,11} (<) {4,9}:{5,8} (>) {2,9}:{10,11} (=) {5,6}:{8,9} (<) (=) (<) {2}:{9} {1}:{ 2} {1,6} :{ 8,11} (<) (=) (>) {2}:{9} {1,2}:{6,7} {8}:{ 11} Fig.22 Where L> R, or L< R, or L= Rmeans L heavier than, or lighter than, or equal to R respectively, and the outputs (x), (a,b,c) and (~) means null, the coins with marks abc,, are heavier ones, and all coins have same weights respectively. The symmetric part of the algorithm has been omitted.

16 The second algorithm to sort 11 coins The difference between the new and the previous one is mainly the Fig.1 to Fig.5 and the third weighing on the direction (=, =) in Fig.0, and the other keep unchanged. Fig.22 (<) Fig.21 (=) {3,4,7,10 } :{1,2,6,11} (<) {5,7,9 } :{6,8,10 } (<) {1,7,8 } :{2,9,10} (<) {1,2,3 } :{4,5,6 } (>) (=) Fig.20 {2,9,11 } :{ 3,4,5} (=) (<) (=) (>) Fig.19 Fig.18 Fig.17 Fig.16 (<) Fig.15 (=) {2,3,6} :{5,8,9} (<) {1,5,7,9}:{2,6,8,10} (=) (>) (=) Fig.14 {1,2,4} :{ 3,7,10} (<) (=) (>) Fig.13 Fig.12 Fig.11 Fig.10 (<) Fig.9 (=) {1,5,6}:{2,3,7} (<) {7,9}:{8,10} (<) {1,7.8}:{ 4,9,10} (>) Fig.8 (=) Fig.7 (<) {5,7,10}:{ 6,8,9} (=) (=) Fig..6 Fig.5 (<) Fig.4 (=) {1,9}:{4,10} (<) {2, 7 }:{ 3, 8} (>) Fig.3 (=) Fig.2 (<) {5} :{ 6} (=) Fig.1 Fig. 0

17 {2}:{3} {4}:{11} (>) (>) {1}:{ 11} (=) {4}:{5} (<) {1,11}:{7,8} (>) {4}:{5} (=) {1}:{ 11} (<) (=) (<) {2}:{11} {2}:{3} {6} :{11 } (<) (=) (>) { 2} :{ 3} (~) {2}:{ 3} Fig.1 {1}:{3} {1}:{ 2} (>) (>) {1}:{ 3} (=) {2,8}:{10,11} (<) {7,9}:{8,10} (>) {3,7}:{9,11} (=) {1}:{ 2} (<) (=) (<) {10}:{11} {9}:{11} {7,10} :{4,11 } (<) (=) (>) {1}:{11} { 1} :{2} {1}:{11} Fig.2 {5}:{ 6} {5}:{6} (>) (>) {5}:{6} (=) {3,5,6}:{2,8,11} (<) {1,4} :{2,8 } (>) {4,8} :{9,11 } (=) {5 } :{6} (<) (=) (<) {9}:{11} {9}:{11} {2,11} :{ 7,10} (<) (=) (>) { 5} :{6} {5}:{6} {9}:{11} Fig.3

18 {6}:{ 11} {6}:{7} (>) (>) {6}:{11} (=) {1,2}:{7,8} (<) {5,10} :{6,11 } (>) {1,4} :{ 7,11} (=) {1 } :{3 } (<) (=) (<) {4}:{6} {10}:{11} {5} :{8} (<) (=) (>) {4}:{ 6} (x) {4}:{6} Fig.4 {1}:{ 11} {4}:{ 11} (>) (>) {1}:{8} (=) {3,6}:{ 4,11} (<) {3,5}:{ 6,10} (>) {5,10}:{ 4,11} (=) { 1} :{2} (<) (=) (<) {1}:{11} {4,10}:{5,11} {5,10} :{6,11 } (<) (=) (>) { 6} :{11} {2}:{4} {2}:{4} Fig.5

19 The third algorithm to sort 11 coins The difference between the third and the first one are in the Figs.3,4,5 and the weighing on the direction (=, =, <) in Fig.0, and the other keep unchanged. Fig.22 (<) Fig.21 (=) {3,4,7,10 } :{1,2,6,11} (<) {5,7,9 } :{6,8,10 } (<) {1,7,8 } :{2,9,10} (<) {1,2,3 } :{4,5,6 } (>) (=) Fig.20 {2,9,11 } :{ 3,4,5} (=) (<) (=) (>) Fig.19 Fig.18 Fig.17 Fig.16 (<) Fig.15 (=) {2,3,6} :{5,8,9} (<) {1,5,7,9}:{2,6,8,10} (=) (>) (=) Fig.14 {1,2,4} :{ 3,7,10} (<) (=) (>) Fig.13 Fig.12 Fig.11 Fig.10 (<) Fig.9 (=) {1,5,6}:{2,3,7} (<) {7,9}:{8,10} (<) {1,7.8}:{ 4,9,10} (>) Fig.8 (=) Fig.7 (<) {5,7,10}:{ 6,8,9} (=) (=) Fig..6 Fig.5 (<) Fig.4 (=) {5,9}:{6,10} (<) {1,2,7}:{3,4,8} (>) Fig.3 (=) Fig.2 (<) {1,3} :{2,4} (=) Fig.1 Fig. 0

20 {5,6}:{ 10,11} {5,6}:{8,9} (>) (>) {1,3}:{2,8} (=) {3,10}:{4,11} (<) {1,4,9}:{3,7,11} (>) {3,11}:{7,9} (=) {2}:{11} (<) (=) (<) {5}:{7} {4,5}:{6,8} {1} :{2} (<) (=) (>) {6}:{7} {5}:{7} {5}:{7} Fig.3 {10}:{ 11} {10}:{11} (>) (>) {8}:{11} (=) {5}:{6} (<) {1}:{2} (>) {5}:{6} (=) {8}:{11} (<) (=) (<) {9}:{11} {9}:{11} {4,7} :{10,11} (<) (=) (>) {10}:{11} {4}:{6} {8}:{11} Fig.4 {5,6}:{10,11} {6,7}:{5,8} (>) (>) {1,3}:{2,8} (=) {3,9}:{4,11} (<) {1,4,10}:{3,7,11} (>) {3,11}:{7,10} (=) {2}:{11} (<) (=) (<) {6}:{7} {4,5}:{6,7} {1} :{2 } (<) (=) (>) { 5} :{7} {6}:{7} {6}:{7} Fig.5

21 Algorithm for Lemma 2 (1) b. b. y 2 3 (<) b. b (=) : bb b y (<) b : b b y (<) xb : yb 1 2 (>) b (=) (>) 2 b. y (<) : b b b : y (=) 1 (=) (>) (<) (=) (>) bbb... y b. y y bb.. y bbb (<) bb. (=) : y b (<) xb 3 : bb 1 2 (>) b. y (>) (=) 1 b (<) : b b y: b 2 (=) (>) (>) (=) (<) b. b. x bb.. y x. b Fig. A Δ= {, x, y}, B = { b1, b2, b3}. ~ := Null

22 Algorithm for Lemma 2 (2) b. y 3 (<) b. b. x. y (=) x : y (<) bb : bx (<) xb : yb (>) (=) (>) b. b. y (<) x : b bb : b x (=) (>) (<) (=) (>) (=) y b.. x y b. y bb.. y bb.. b. y (<). bb. xy. (=) : b b (<) xb 3 : bb 1 2 (>) b. y (=) (>) 1 bb.. y (<) x : y y: b (=) (>) (>) (=) (<) bb.. b. xy. b. x x. y b.. x y b. b. x Fig. B Δ= {, x yxy,.}, B = { b1, b2, b3}.

23 Algorithm for Lemma 2 (3) yb e : zb b y yb.. b (~) b bb. bbb (>) (=) (<) (>) (=) (<) (>) (=) (<) y: b be: bb yb. (>) (>) yb.. b (>) (>) yzb.. (=) y: z (=) yb : b b b : b bb : yz (=) yb : b b (=) yzb... b. b (<) (<) (~) (<) (>) (=) (<) (<) b 2 (~) (>) b : b : b b (<) yzb.. 2 (=) (<) : 3 y b z: b y: b (>) (=) 1 3 yb.. b b (>) (=) (<) (>) (=) (<) (>) (=) (<) yzb.. yzb... b yb yzb b bb. zy. yb.. b. b yb yzb b b. b 2 3 Fig. C Δ= {, yyz,. }, B = { b1, b2, b3}, e is a normal coin.

24 Algorithm g (6) = 4 in Lemma (<) 5.6 (=) { 2} :{3} (<) { 4} :{6} (<) {1,4}:{2,5} (<) {1,2,3}:{4,5,6} (>) (=) (>) {1,5 } :{ 2,3} (=) (<) (=) (>) (<) 6 (=) {2} :{3} (<) {1,5}:{3,6} (=) (>) (=) (>) (<) {3}:{6} {1} :{4} (=) (>) (<) (=) (>) ~ (<) (=) {1} :{2} (<) {3} :{6 } (<) {1,4 } :{ 2,5} (>) ~ (=) (>) ~ (<) {2}:{3} {2}:{3} (=) (=) (>) (<) (=) (>) ~ (<) 3.6 (=) {1}:{2} (<) {1,2,}:{3,6} (>) (=) (>) {1}:{2} (<) (=) (>) Fig. D

25 The completed list of weighings of the first algorithm to sort 11 coins The 1-th weighing w( ) = {1,2,3}:{4,5,6} The 2-th weighing w(0) = {1,7,8}:{4,9,10} w(1) = {1,7,8}:{2,9,10} w(2) = {1,7,8}:{2,9,10} The 3-th weighing w(0,0) = {1,2,7}:{3,4,8} w(1,0) = {1,5,7,9}:{2,6,8,10} w(2,0) = {1,5,7,9}:{2,6,8,10} w(0,1) = {7,9}:{8,10} w(1,1) = {5,7,9}:{6,8,10} w(2,1) = {5,7,9}:{6,8,10} w(0,2) = {7,9}:{8,10} w(1,2) = {5,7,9}:{6,8,10} w(2,2) = {5,7,9}:{6,8,10} The 4-th weighing w(0,0,0) = {1,3}:{2,4} w(1,0,0) = {1,2,4}:{3,7,10} w(2,0,0) = {1,2,4}:{3,7,10} w(0,1,0) = {5,7,10}:{6,8,9} w(1,1,0) = {2,9,11}:{3,4,5} w(2,1,0) = {1,7,11}:{3,4,5} w(0,2,0) = {5,7,10}:{6,8,9} w(1,2,0) = {1,7,11}:{3,4,5} w(2,2,0) = {2,9,11}:{3,4,5} w(0,0,1) = {1,3,4}:{8,9,11} w(1,0,1) = {2,3,6}:{5,8,9} w(2,0,1) = {1,3,5}:{6,8,9} w(0,1,1) = {1,5,6}:{2,3,7} w(1,1,1) = {3,4,7,10}:{1,2,6,11} w(2,1,1) = {3,4,7,10}:{1,2,5,11} w(0,2,1) = {1,5,6}:{2,3,8} w(1,2,1) = {3,4,8,9}:{1,2,6,11} w(2,2,1) = {3,4,8,9}:{1,2,5,11} w(0,0,2) = {1,3,4}:{8,9,11} w(1,0,2) = {1,3,5}:{6,8,9} w(2,0,2) = {2,3,6}:{5,8,9} w(0,1,2) = {1,5,6}:{2,3,8} w(1,1,2) = {3,4,8,9}:{1,2,5,11} w(2,1,2) = {3,4,8,9}:{1,2,6,11} w(0,2,2) = {1,5,6}:{2,3,7} w(1,2,2) = {3,4,7,10}:{1,2,5,11} w(2,2,2) = {3,4,7,10}:{1,2,6,11} The 5-th weighing w(0,0,0,0) = {1,2}:{10,11} w(1,0,0,0) = {1}:{2} w(2,0,0,0) = {1}:{2} w(0,1,0,0) = {2}:{3} w(1,1,0,0) = {2,6,8}:{3,5,11} w(2,1,0,0) = {1,6,10}:{3,5,11} w(0,2,0,0) = {2}:{3} w(1,2,0,0) = {1,6,10}:{3,5,11} w(2,2,0,0) = {2,6,8}:{3,5,11} w(0,0,1,0) = {5}:{6} w(1,0,1,0) = {2,5,9}:{3,4,7} w(2,0,1,0) = {1,6,8}:{3,4,10} w(0,1,1,0) = {2}:{3} w(1,1,1,0) = {1,2}:{7,11} w(2,1,1,0) = {1,2}:{10,11} w(0,2,1,0) = {2}:{3} w(1,2,1,0) = {1,2}:{9,11} w(2,2,1,0) = {1,2}:{8,11} w(0,0,2,0) = {5}:{6} w(1,0,2,0) = {1,6,8}:{3,4,10} w(2,0,2,0) = {2,5,9}:{3,4,7} w(0,1,2,0) = {2}:{3} w(1,1,2,0) = {1,2}:{8,11} w(2,1,2,0) = {1,2}:{9,11} w(0,2,2,0) = {2}:{3} w(1,2,2,0) = {1,2}:{10,11} w(2,2,2,0) = {1,2}:{7,11} w(0,0,0,1) = {1}:{7} w(1,0,0,1) = {3,7,10}:{4,5,6} w(2,0,0,1) = {1,2}:{8,9} w(0,1,0,1) = {2,5}:{3,7} w(1,1,0,1) = {1,5,9}:{2,4,11} w(2,1,0,1) = {2,4,11}:{3,5,8} w(0,2,0,1) = {2,6}:{3,8} w(1,2,0,1) = {2,5,7}:{1,4,11} w(2,2,0,1) = {1,4,11}:{3,5,10} w(0,0,1,1) = {3,5}:{6,9} w(1,0,1,1) = {1,6,11}:{3,7,10} w(2,0,1,1) = {3,6,7}:{8,10,11} w(0,1,1,1) = {1,10}:{8,9} w(1,1,1,1) = {4,9}:{5,8} w(2,1,1,1) = {2,5,6}:{4,10,11} w(0,2,1,1) = {2}:{3} w(1,2,1,1) = {4,7}:{5,10} w(2,2,1,1) = {1,5,6}:{4,8,11} w(0,0,2,1) = {5}:{6} w(1,0,2,1) = {2,5,11}:{3,7,10} w(2,0,2,1) = {3,5,10}:{7,9,11} w(0,1,2,1) = {1,9}:{7,10} w(1,1,2,1) = {4,10}:{6,7} w(2,1,2,1) = {2,5,6}:{4,9,11} w(0,2,2,1) = {2}:{3} w(1,2,2,1) = {4,8}:{6,9} w(2,2,2,1) = {1,5,6}:{4,7,11} w(0,0,0,2) = {1}:{7} w(1,0,0,2) = {1,2}:{8,9} w(2,0,0,2) = {3,7,10}:{4,5,6} w(0,1,0,2) = {2,6}:{3,8} w(1,1,0,2) = {1,4,11}:{3,5,10} w(2,1,0,2) = {2,5,7}:{1,4,11} w(0,2,0,2) = {2,5}:{3,7} w(1,2,0,2) = {2,4,11}:{3,5,8} w(2,2,0,2) = {1,5,9}:{2,4,11} w(0,0,1,2) = {5}:{6} w(1,0,1,2) = {3,5,10}:{7,9,11} w(2,0,1,2) = {2,5,11}:{3,7,10}

26 w(0,1,1,2) = {2}:{3} w(1,1,1,2) = {1,5,6}:{4,7,11} w(2,1,1,2) = {4,8}:{6,9} w(0,2,1,2) = {1,9}:{7,10} w(1,2,1,2) = {2,5,6}:{4,9,11} w(2,2,1,2) = {4,10}:{6,7} w(0,0,2,2) = {3,5}:{6,9} w(1,0,2,2) = {3,6,7}:{8,10,11} w(2,0,2,2) = {1,6,11}:{3,7,10} w(0,1,2,2) = {2}:{3} w(1,1,2,2) = {1,5,6}:{4,8,11} w(2,1,2,2) = {4,7}:{5,10} w(0,2,2,2) = {1,10}:{8,9} w(1,2,2,2) = {2,5,6}:{4,10,11} w(2,2,2,2) = {4,9}:{5,8} The 6-th weighing w(0,0,0,0,0) = {1}:{2} w(1,0,0,0,0) = {3,4}:{9,11} w(2,0,0,0,0) = {3,4}:{9,11} w(0,1,0,0,0) = {1,11}:{2,3} w(1,1,0,0,0) = {4}:{9} w(2,1,0,0,0) = {4}:{7} w(0,2,0,0,0) = {1,11}:{2,3} w(1,2,0,0,0) = {4}:{7} w(2,2,0,0,0) = {4}:{9} w(0,0,1,0,0) = {9}:{11} w(1,0,1,0,0) = {2,9}:{3,11} w(2,0,1,0,0) = {1,8}:{3,11} w(0,1,1,0,0) = {5,6}:{9,11} w(1,1,1,0,0) = {2,8}:{7,10} w(2,1,1,0,0) = {1,9}:{7,10} w(0,2,1,0,0) = {5,6}:{10,11} w(1,2,1,0,0) = {1,10}:{8,9} w(2,2,1,0,0) = {2,7}:{8,9} w(0,0,2,0,0) = {9}:{11} w(1,0,2,0,0) = {1,8}:{3,11} w(2,0,2,0,0) = {2,9}:{3,11} w(0,1,2,0,0) = {5,6}:{10,11} w(1,1,2,0,0) = {2,7}:{8,9} w(2,1,2,0,0) = {1,10}:{8,9} w(0,2,2,0,0) = {5,6}:{9,11} w(1,2,2,0,0) = {1,9}:{7,10} w(2,2,2,0,0) = {2,8}:{7,10} w(0,0,0,1,0) = {5,11}:{6,9} w(1,0,0,1,0) = {3,4}:{9,11} w(2,0,0,1,0) = {1,2}:{10,11} w(0,1,0,1,0) = {1,10}:{8,11} w(1,1,0,1,0) = {1,4}:{2,6} w(2,1,0,1,0) = {7,9}:{10,11} w(0,2,0,1,0) = {1,9}:{7,11} w(1,2,0,1,0) = {2,4}:{1,6} w(2,2,0,1,0) = {7,9}:{8,11} w(0,0,1,1,0) = {9}:{10} w(1,0,1,1,0) = {5}:{6} w(2,0,1,1,0) = {1,4}:{7,11} w(0,1,1,1,0) = {1,3,11}:{2,5,6} w(1,1,1,1,0) = {1,6}:{8,11} w(2,1,1,1,0) = {8,9}:{5,11} w(0,2,1,1,0) = {5,6}:{10,11} w(1,2,1,1,0) = {2,6}:{10,11} w(2,2,1,1,0) = {7,10}:{5,11} w(0,0,2,1,0) = {2}:{7} w(1,0,2,1,0) = {6}:{5} w(2,0,2,1,0) = {2,4}:{10,11} w(0,1,2,1,0) = {1,3,11}:{2,5,6} w(1,1,2,1,0) = {1,5}:{7,11} w(2,1,2,1,0) = {7,10}:{6,11} w(0,2,2,1,0) = {5,6}:{9,11} w(1,2,2,1,0) = {2,5}:{9,11} w(2,2,2,1,0) = {8,9}:{6,11} w(0,0,0,2,0) = {5,11}:{6,9} w(1,0,0,2,0) = {1,2}:{10,11} w(2,0,0,2,0) = {3,4}:{9,11} w(0,1,0,2,0) = {1,9}:{7,11} w(1,1,0,2,0) = {7,9}:{8,11} w(2,1,0,2,0) = {2,4}:{1,6} w(0,2,0,2,0) = {1,10}:{8,11} w(1,2,0,2,0) = {7,9}:{10,11} w(2,2,0,2,0) = {1,4}:{2,6} w(0,0,1,2,0) = {2}:{7} w(1,0,1,2,0) = {2,4}:{10,11} w(2,0,1,2,0) = {6}:{5} w(0,1,1,2,0) = {5,6}:{9,11} w(1,1,1,2,0) = {8,9}:{6,11} w(2,1,1,2,0) = {2,5}:{9,11} w(0,2,1,2,0) = {1,3,11}:{2,5,6} w(1,2,1,2,0) = {7,10}:{6,11} w(2,2,1,2,0) = {1,5}:{7,11} w(0,0,2,2,0) = {9}:{10} w(1,0,2,2,0) = {1,4}:{7,11} w(2,0,2,2,0) = {5}:{6} w(0,1,2,2,0) = {5,6}:{10,11} w(1,1,2,2,0) = {7,10}:{5,11} w(2,1,2,2,0) = {2,6}:{10,11} w(0,2,2,2,0) = {1,3,11}:{2,5,6} w(1,2,2,2,0) = {8,9}:{5,11} w(2,2,2,2,0) = {1,6}:{8,11} w(0,0,0,0,1) = {10}:{11} w(1,0,0,0,1) = {3,4}:{9,11} w(2,0,0,0,1) = {3,4}:{9,11} w(0,1,0,0,1) = {3,11}:{9,10} w(1,1,0,0,1) = {1,6}:{5,8} w(2,1,0,0,1) = {2,6}:{7,10} w(0,2,0,0,1) = {2,11}:{9,10} w(1,2,0,0,1) = {2,6}:{5,10} w(2,2,0,0,1) = {1,6}:{8,9} w(0,0,1,0,1) = {9}:{10} w(1,0,1,0,1) = {5,11}:{7,9} w(2,0,1,0,1) = {2,4}:{10,11} w(0,1,1,0,1) = {5,9}:{6,11} w(1,1,1,0,1) = {2,4}:{5,9} w(2,1,1,0,1) = {6}:{8} w(0,2,1,0,1) = {5,10}:{6,11} w(1,2,1,0,1) = {1,4}:{5,7} w(2,2,1,0,1) = {6}:{10} w(0,0,2,0,1) = {9}:{10} w(1,0,2,0,1) = {6,11}:{8,10} w(2,0,2,0,1) = {1,4}:{7,11} w(0,1,2,0,1) = {5,10}:{6,11} w(1,1,2,0,1) = {2,4}:{6,10} w(2,1,2,0,1) = {5}:{7} w(0,2,2,0,1) = {5,9}:{6,11} w(1,2,2,0,1) = {1,4}:{6,8} w(2,2,2,0,1) = {5}:{9} w(0,0,0,1,1) = {3,5}:{1,11} w(1,0,0,1,1) = {3,8}:{4,11} w(2,0,0,1,1) = {1,10}:{2,11} w(0,1,0,1,1) = {1,11}:{2,8} w(1,1,0,1,1) = {3,6}:{7,10} w(2,1,0,1,1) = {5,6}:{9,11}

27 w(0,2,0,1,1) = {1,5}:{6,11} w(1,2,0,1,1) = {3,6}:{8,9} w(2,2,0,1,1) = {5,6}:{7,11} w(0,0,1,1,1) = {3,6}:{7,9} w(1,0,1,1,1) = {1,5}:{7,11} w(2,0,1,1,1) = {1,9}:{7,11} w(0,1,1,1,1) = {1,3,11}:{2,5,6} w(1,1,1,1,1) = {1,2}:{4,11} w(2,1,1,1,1) = {1,4}:{3,8} w(0,2,1,1,1) = {5,10}:{6,11} w(1,2,1,1,1) = {1,2}:{4,11} w(2,2,1,1,1) = {2,4}:{3,10} w(0,0,2,1,1) = {1,9}:{8,11} w(1,0,2,1,1) = {2,6}:{10,11} w(2,0,2,1,1) = {2,8}:{10,11} w(0,1,2,1,1) = {1,3,11}:{2,5,6} w(1,1,2,1,1) = {1,2}:{4,11} w(2,1,2,1,1) = {1,4}:{3,7} w(0,2,2,1,1) = {5,9}:{6,11} w(1,2,2,1,1) = {1,2}:{4,11} w(2,2,2,1,1) = {2,4}:{3,9} w(0,0,0,2,1) = {5,11}:{6,9} w(1,0,0,2,1) = {1,5}:{7,11} w(2,0,0,2,1) = {3,5}:{9,11} w(0,1,0,2,1) = {1,11}:{2,7} w(1,1,0,2,1) = {2,9}:{10,11} w(2,1,0,2,1) = {2,4}:{7,10} w(0,2,0,2,1) = {1,6}:{5,11} w(1,2,0,2,1) = {1,7}:{8,11} w(2,2,0,2,1) = {1,4}:{8,9} w(0,0,1,2,1) = {1,9}:{8,11} w(1,0,1,2,1) = {2}:{10} w(2,0,1,2,1) = {6,7}:{1,11} w(0,1,1,2,1) = {5,9}:{6,11} w(1,1,1,2,1) = {8,9}:{3,11} w(2,1,1,2,1) = {1,8}:{7,11} w(0,2,1,2,1) = {1,3,11}:{2,5,6} w(1,2,1,2,1) = {7,10}:{3,11} w(2,2,1,2,1) = {2,10}:{9,11} w(0,0,2,2,1) = {9}:{11} w(1,0,2,2,1) = {1}:{7} w(2,0,2,2,1) = {5,10}:{2,11} w(0,1,2,2,1) = {5,10}:{6,11} w(1,1,2,2,1) = {7,10}:{3,11} w(2,1,2,2,1) = {1,7}:{8,11} w(0,2,2,2,1) = {1,3,11}:{2,5,6} w(1,2,2,2,1) = {8,9}:{3,11} w(2,2,2,2,1) = {2,9}:{10,11} w(0,0,0,0,2) = {10}:{11} w(1,0,0,0,2) = {3,4}:{9,11} w(2,0,0,0,2) = {3,4}:{9,11} w(0,1,0,0,2) = {2,11}:{9,10} w(1,1,0,0,2) = {1,6}:{8,9} w(2,1,0,0,2) = {2,6}:{5,10} w(0,2,0,0,2) = {3,11}:{9,10} w(1,2,0,0,2) = {2,6}:{7,10} w(2,2,0,0,2) = {1,6}:{5,8} w(0,0,1,0,2) = {9}:{10} w(1,0,1,0,2) = {1,4}:{7,11} w(2,0,1,0,2) = {6,11}:{8,10} w(0,1,1,0,2) = {5,9}:{6,11} w(1,1,1,0,2) = {5}:{9} w(2,1,1,0,2) = {1,4}:{6,8} w(0,2,1,0,2) = {5,10}:{6,11} w(1,2,1,0,2) = {5}:{7} w(2,2,1,0,2) = {2,4}:{6,10} w(0,0,2,0,2) = {9}:{10} w(1,0,2,0,2) = {2,4}:{10,11} w(2,0,2,0,2) = {5,11}:{7,9} w(0,1,2,0,2) = {5,10}:{6,11} w(1,1,2,0,2) = {6}:{10} w(2,1,2,0,2) = {1,4}:{5,7} w(0,2,2,0,2) = {5,9}:{6,11} w(1,2,2,0,2) = {6}:{8} w(2,2,2,0,2) = {2,4}:{5,9} w(0,0,0,1,2) = {5,11}:{6,9} w(1,0,0,1,2) = {3,5}:{9,11} w(2,0,0,1,2) = {1,5}:{7,11} w(0,1,0,1,2) = {1,6}:{5,11} w(1,1,0,1,2) = {1,4}:{8,9} w(2,1,0,1,2) = {1,7}:{8,11} w(0,2,0,1,2) = {1,11}:{2,7} w(1,2,0,1,2) = {2,4}:{7,10} w(2,2,0,1,2) = {2,9}:{10,11} w(0,0,1,1,2) = {9}:{11} w(1,0,1,1,2) = {5,10}:{2,11} w(2,0,1,1,2) = {1}:{7} w(0,1,1,1,2) = {1,3,11}:{2,5,6} w(1,1,1,1,2) = {2,9}:{10,11} w(2,1,1,1,2) = {8,9}:{3,11} w(0,2,1,1,2) = {5,10}:{6,11} w(1,2,1,1,2) = {1,7}:{8,11} w(2,2,1,1,2) = {7,10}:{3,11} w(0,0,2,1,2) = {1,9}:{8,11} w(1,0,2,1,2) = {6,7}:{1,11} w(2,0,2,1,2) = {2}:{10} w(0,1,2,1,2) = {1,3,11}:{2,5,6} w(1,1,2,1,2) = {2,10}:{9,11} w(2,1,2,1,2) = {7,10}:{3,11} w(0,2,2,1,2) = {5,9}:{6,11} w(1,2,2,1,2) = {1,8}:{7,11} w(2,2,2,1,2) = {8,9}:{3,11} w(0,0,0,2,2) = {3,5}:{1,11} w(1,0,0,2,2) = {1,10}:{2,11} w(2,0,0,2,2) = {3,8}:{4,11} w(0,1,0,2,2) = {1,5}:{6,11} w(1,1,0,2,2) = {5,6}:{7,11} w(2,1,0,2,2) = {3,6}:{8,9} w(0,2,0,2,2) = {1,11}:{2,8} w(1,2,0,2,2) = {5,6}:{9,11} w(2,2,0,2,2) = {3,6}:{7,10} w(0,0,1,2,2) = {1,9}:{8,11} w(1,0,1,2,2) = {2,8}:{10,11} w(2,0,1,2,2) = {2,6}:{10,11} w(0,1,1,2,2) = {5,9}:{6,11} w(1,1,1,2,2) = {2,4}:{3,9} w(2,1,1,2,2) = {1,2}:{4,11} w(0,2,1,2,2) = {1,3,11}:{2,5,6} w(1,2,1,2,2) = {1,4}:{3,7} w(2,2,1,2,2) = {1,2}:{4,11} w(0,0,2,2,2) = {3,6}:{7,9} w(1,0,2,2,2) = {1,9}:{7,11} w(2,0,2,2,2) = {1,5}:{7,11} w(0,1,2,2,2) = {5,10}:{6,11} w(1,1,2,2,2) = {2,4}:{3,10} w(2,1,2,2,2) = {1,2}:{4,11} w(0,2,2,2,2) = {1,3,11}:{2,5,6} w(1,2,2,2,2) = {1,4}:{3,8} w(2,2,2,2,2) = {1,2}:{4,11} The 7-th weighing

28 w(0,0,0,0,0,0) = {}:{} w(1,0,0,0,0,0) = {6}:{8} w(2,0,0,0,0,0) = {6}:{8} w(0,1,0,0,0,0) = {6}:{9} w(1,1,0,0,0,0) = {10}:{11} w(2,1,0,0,0,0) = {8}:{11} w(0,2,0,0,0,0) = {6}:{9} w(1,2,0,0,0,0) = {8}:{11} w(2,2,0,0,0,0) = {10}:{11} w(0,0,1,0,0,0) = {9}:{10} w(1,0,1,0,0,0) = {10}:{11} w(2,0,1,0,0,0) = {7}:{11} w(0,1,1,0,0,0) = {9}:{10} w(1,1,1,0,0,0) = {5}:{9} w(2,1,1,0,0,0) = {6}:{8} w(0,2,1,0,0,0) = {10}:{9} w(1,2,1,0,0,0) = {5}:{7} w(2,2,1,0,0,0) = {6}:{10} w(0,0,2,0,0,0) = {9}:{10} w(1,0,2,0,0,0) = {7}:{11} w(2,0,2,0,0,0) = {10}:{11} w(0,1,2,0,0,0) = {10}:{9} w(1,1,2,0,0,0) = {6}:{10} w(2,1,2,0,0,0) = {5}:{7} w(0,2,2,0,0,0) = {9}:{10} w(1,2,2,0,0,0) = {6}:{8} w(2,2,2,0,0,0) = {5}:{9} w(0,0,0,1,0,0) = {9}:{10} w(1,0,0,1,0,0) = {3}:{5} w(2,0,0,1,0,0) = {3,4}:{6,7} w(0,1,0,1,0,0) = {6}:{9} w(1,1,0,1,0,0) = {5}:{7} w(2,1,0,1,0,0) = {2}:{9} w(0,2,0,1,0,0) = {5}:{10} w(1,2,0,1,0,0) = {5}:{9} w(2,2,0,1,0,0) = {1}:{7} w(0,0,1,1,0,0) = {1}:{2} w(1,0,1,1,0,0) = {4}:{6} w(2,0,1,1,0,0) = {7}:{11} w(0,1,1,1,0,0) = {5}:{6} w(1,1,1,1,0,0) = {1,2}:{6,7} w(2,1,1,1,0,0) = {1,4}:{3,6} w(0,2,1,1,0,0) = {10}:{9} w(1,2,1,1,0,0) = {1,2}:{6,9} w(2,2,1,1,0,0) = {2,4}:{3,6} w(0,0,2,1,0,0) = {1}:{3} w(1,0,2,1,0,0) = {4}:{5} w(2,0,2,1,0,0) = {10}:{11} w(0,1,2,1,0,0) = {5}:{6} w(1,1,2,1,0,0) = {1,2}:{5,8} w(2,1,2,1,0,0) = {1,4}:{3,5} w(0,2,2,1,0,0) = {9}:{10} w(1,2,2,1,0,0) = {1,2}:{5,10} w(2,2,2,1,0,0) = {2,4}:{3,5} w(0,0,0,2,0,0) = {9}:{10} w(1,0,0,2,0,0) = {3,4}:{6,7} w(2,0,0,2,0,0) = {3}:{5} w(0,1,0,2,0,0) = {5}:{10} w(1,1,0,2,0,0) = {1}:{7} w(2,1,0,2,0,0) = {5}:{9} w(0,2,0,2,0,0) = {6}:{9} w(1,2,0,2,0,0) = {2}:{9} w(2,2,0,2,0,0) = {5}:{7} w(0,0,1,2,0,0) = {1}:{3} w(1,0,1,2,0,0) = {10}:{11} w(2,0,1,2,0,0) = {4}:{5} w(0,1,1,2,0,0) = {9}:{10} w(1,1,1,2,0,0) = {2,4}:{3,5} w(2,1,1,2,0,0) = {1,2}:{5,10} w(0,2,1,2,0,0) = {5}:{6} w(1,2,1,2,0,0) = {1,4}:{3,5} w(2,2,1,2,0,0) = {1,2}:{5,8} w(0,0,2,2,0,0) = {1}:{2} w(1,0,2,2,0,0) = {7}:{11} w(2,0,2,2,0,0) = {4}:{6} w(0,1,2,2,0,0) = {10}:{9} w(1,1,2,2,0,0) = {2,4}:{3,6} w(2,1,2,2,0,0) = {1,2}:{6,9} w(0,2,2,2,0,0) = {5}:{6} w(1,2,2,2,0,0) = {1,4}:{3,6} w(2,2,2,2,0,0) = {1,2}:{6,7} w(0,0,0,0,1,0) = {1}:{2} w(1,0,0,0,1,0) = {6}:{8} w(2,0,0,0,1,0) = {10}:{11} w(0,1,0,0,1,0) = {1}:{7} w(1,1,0,0,1,0) = {3}:{4} w(2,1,0,0,1,0) = {1}:{9} w(0,2,0,0,1,0) = {1}:{7} w(1,2,0,0,1,0) = {3}:{4} w(2,2,0,0,1,0) = {2}:{7} w(0,0,1,0,1,0) = {1}:{2} w(1,0,1,0,1,0) = {10}:{11} w(2,0,1,0,1,0) = {1}:{4} w(0,1,1,0,1,0) = {9}:{11} w(1,1,1,0,1,0) = {5}:{8} w(2,1,1,0,1,0) = {5,10}:{8,9} w(0,2,1,0,1,0) = {10}:{11} w(1,2,1,0,1,0) = {5}:{10} w(2,2,1,0,1,0) = {5,8}:{7,10} w(0,0,2,0,1,0) = {1}:{2} w(1,0,2,0,1,0) = {7}:{11} w(2,0,2,0,1,0) = {2}:{4} w(0,1,2,0,1,0) = {10}:{11} w(1,1,2,0,1,0) = {6}:{7} w(2,1,2,0,1,0) = {6,9}:{7,10} w(0,2,2,0,1,0) = {9}:{11} w(1,2,2,0,1,0) = {6}:{9} w(2,2,2,0,1,0) = {6,7}:{8,9} w(0,0,0,1,1,0) = {9}:{10} w(1,0,0,1,1,0) = {3}:{4} w(2,0,0,1,1,0) = {2}:{11} w(0,1,0,1,1,0) = {6}:{8} w(1,1,0,1,1,0) = {2}:{11} w(2,1,0,1,1,0) = {1,4}:{5,6} w(0,2,0,1,1,0) = {5}:{7} w(1,2,0,1,1,0) = {1}:{11} w(2,2,0,1,1,0) = {2,4}:{5,6} w(0,0,1,1,1,0) = {10}:{11} w(1,0,1,1,1,0) = {2}:{3} w(2,0,1,1,1,0) = {2,3}:{9,10} w(0,1,1,1,1,0) = {5}:{6} w(1,1,1,1,1,0) = {6,7}:{8,9} w(2,1,1,1,1,0) = {3}:{8} w(0,2,1,1,1,0) = {10}:{11} w(1,2,1,1,1,0) = {6,9}:{7,10} w(2,2,1,1,1,0) = {3}:{10} w(0,0,2,1,1,0) = {10}:{11} w(1,0,2,1,1,0) = {1}:{3} w(2,0,2,1,1,0) = {1,3}:{7,8} w(0,1,2,1,1,0) = {5}:{6} w(1,1,2,1,1,0) = {5,8}:{7,10} w(2,1,2,1,1,0) = {3}:{7}

29 w(0,2,2,1,1,0) = {9}:{11} w(1,2,2,1,1,0) = {5,10}:{8,9} w(2,2,2,1,1,0) = {3}:{9} w(0,0,0,2,1,0) = {5}:{6} w(1,0,0,2,1,0) = {2}:{11} w(2,0,0,2,1,0) = {3}:{4} w(0,1,0,2,1,0) = {5}:{7} w(1,1,0,2,1,0) = {2}:{7} w(2,1,0,2,1,0) = {1,5}:{3,4} w(0,2,0,2,1,0) = {6}:{8} w(1,2,0,2,1,0) = {1}:{9} w(2,2,0,2,1,0) = {2,5}:{3,4} w(0,0,1,2,1,0) = {10}:{11} w(1,0,1,2,1,0) = {2,4}:{5,6} w(2,0,1,2,1,0) = {2,4}:{5,6} w(0,1,1,2,1,0) = {9}:{11} w(1,1,1,2,1,0) = {2}:{3} w(2,1,1,2,1,0) = {5,6}:{8,9} w(0,2,1,2,1,0) = {5}:{6} w(1,2,1,2,1,0) = {1}:{3} w(2,2,1,2,1,0) = {5,6}:{7,10} w(0,0,2,2,1,0) = {2}:{10} w(1,0,2,2,1,0) = {1,4}:{5,6} w(2,0,2,2,1,0) = {1,4}:{5,6} w(0,1,2,2,1,0) = {10}:{11} w(1,1,2,2,1,0) = {2}:{3} w(2,1,2,2,1,0) = {5,6}:{7,10} w(0,2,2,2,1,0) = {5}:{6} w(1,2,2,2,1,0) = {1}:{3} w(2,2,2,2,1,0) = {5,6}:{8,9} w(0,0,0,0,2,0) = {1}:{2} w(1,0,0,0,2,0) = {10}:{11} w(2,0,0,0,2,0) = {6}:{8} w(0,1,0,0,2,0) = {1}:{7} w(1,1,0,0,2,0) = {2}:{7} w(2,1,0,0,2,0) = {3}:{4} w(0,2,0,0,2,0) = {1}:{7} w(1,2,0,0,2,0) = {1}:{9} w(2,2,0,0,2,0) = {3}:{4} w(0,0,1,0,2,0) = {1}:{2} w(1,0,1,0,2,0) = {2}:{4} w(2,0,1,0,2,0) = {7}:{11} w(0,1,1,0,2,0) = {9}:{11} w(1,1,1,0,2,0) = {6,7}:{8,9} w(2,1,1,0,2,0) = {6}:{9} w(0,2,1,0,2,0) = {10}:{11} w(1,2,1,0,2,0) = {6,9}:{7,10} w(2,2,1,0,2,0) = {6}:{7} w(0,0,2,0,2,0) = {1}:{2} w(1,0,2,0,2,0) = {1}:{4} w(2,0,2,0,2,0) = {10}:{11} w(0,1,2,0,2,0) = {10}:{11} w(1,1,2,0,2,0) = {5,8}:{7,10} w(2,1,2,0,2,0) = {5}:{10} w(0,2,2,0,2,0) = {9}:{11} w(1,2,2,0,2,0) = {5,10}:{8,9} w(2,2,2,0,2,0) = {5}:{8} w(0,0,0,1,2,0) = {5}:{6} w(1,0,0,1,2,0) = {3}:{4} w(2,0,0,1,2,0) = {2}:{11} w(0,1,0,1,2,0) = {6}:{8} w(1,1,0,1,2,0) = {2,5}:{3,4} w(2,1,0,1,2,0) = {1}:{9} w(0,2,0,1,2,0) = {5}:{7} w(1,2,0,1,2,0) = {1,5}:{3,4} w(2,2,0,1,2,0) = {2}:{7} w(0,0,1,1,2,0) = {2}:{10} w(1,0,1,1,2,0) = {1,4}:{5,6} w(2,0,1,1,2,0) = {1,4}:{5,6} w(0,1,1,1,2,0) = {5}:{6} w(1,1,1,1,2,0) = {5,6}:{8,9} w(2,1,1,1,2,0) = {1}:{3} w(0,2,1,1,2,0) = {10}:{11} w(1,2,1,1,2,0) = {5,6}:{7,10} w(2,2,1,1,2,0) = {2}:{3} w(0,0,2,1,2,0) = {10}:{11} w(1,0,2,1,2,0) = {2,4}:{5,6} w(2,0,2,1,2,0) = {2,4}:{5,6} w(0,1,2,1,2,0) = {5}:{6} w(1,1,2,1,2,0) = {5,6}:{7,10} w(2,1,2,1,2,0) = {1}:{3} w(0,2,2,1,2,0) = {9}:{11} w(1,2,2,1,2,0) = {5,6}:{8,9} w(2,2,2,1,2,0) = {2}:{3} w(0,0,0,2,2,0) = {9}:{10} w(1,0,0,2,2,0) = {2}:{11} w(2,0,0,2,2,0) = {3}:{4} w(0,1,0,2,2,0) = {5}:{7} w(1,1,0,2,2,0) = {2,4}:{5,6} w(2,1,0,2,2,0) = {1}:{11} w(0,2,0,2,2,0) = {6}:{8} w(1,2,0,2,2,0) = {1,4}:{5,6} w(2,2,0,2,2,0) = {2}:{11} w(0,0,1,2,2,0) = {10}:{11} w(1,0,1,2,2,0) = {1,3}:{7,8} w(2,0,1,2,2,0) = {1}:{3} w(0,1,1,2,2,0) = {9}:{11} w(1,1,1,2,2,0) = {3}:{9} w(2,1,1,2,2,0) = {5,10}:{8,9} w(0,2,1,2,2,0) = {5}:{6} w(1,2,1,2,2,0) = {3}:{7} w(2,2,1,2,2,0) = {5,8}:{7,10} w(0,0,2,2,2,0) = {10}:{11} w(1,0,2,2,2,0) = {2,3}:{9,10} w(2,0,2,2,2,0) = {2}:{3} w(0,1,2,2,2,0) = {10}:{11} w(1,1,2,2,2,0) = {3}:{10} w(2,1,2,2,2,0) = {6,9}:{7,10} w(0,2,2,2,2,0) = {5}:{6} w(1,2,2,2,2,0) = {3}:{8} w(2,2,2,2,2,0) = {6,7}:{8,9} w(0,0,0,0,0,1) = {10}:{11} w(1,0,0,0,0,1) = {8}:{11} w(2,0,0,0,0,1) = {8}:{11} w(0,1,0,0,0,1) = {1}:{11} w(1,1,0,0,0,1) = {1}:{8} w(2,1,0,0,0,1) = {1}:{9} w(0,2,0,0,0,1) = {1}:{11} w(1,2,0,0,0,1) = {2}:{10} w(2,2,0,0,0,1) = {2}:{7} w(0,0,1,0,0,1) = {3}:{10} w(1,0,1,0,0,1) = {3}:{11} w(2,0,1,0,0,1) = {7}:{11} w(0,1,1,0,0,1) = {9}:{11} w(1,1,1,0,0,1) = {5}:{9} w(2,1,1,0,0,1) = {3,5}:{6,8} w(0,2,1,0,0,1) = {10}:{11} w(1,2,1,0,0,1) = {5}:{7} w(2,2,1,0,0,1) = {3,5}:{6,10} w(0,0,2,0,0,1) = {3}:{4} w(1,0,2,0,0,1) = {3}:{11} w(2,0,2,0,0,1) = {10}:{11}

30 w(0,1,2,0,0,1) = {10}:{11} w(1,1,2,0,0,1) = {6}:{10} w(2,1,2,0,0,1) = {3,6}:{5,7} w(0,2,2,0,0,1) = {9}:{11} w(1,2,2,0,0,1) = {6}:{8} w(2,2,2,0,0,1) = {3,6}:{5,9} w(0,0,0,1,0,1) = {10}:{11} w(1,0,0,1,0,1) = {9}:{11} w(2,0,0,1,0,1) = {10}:{11} w(0,1,0,1,0,1) = {8}:{11} w(1,1,0,1,0,1) = {8}:{10} w(2,1,0,1,0,1) = {1}:{11} w(0,2,0,1,0,1) = {2}:{11} w(1,2,0,1,0,1) = {10}:{8} w(2,2,0,1,0,1) = {2}:{11} w(0,0,1,1,0,1) = {2,10}:{3,11} w(1,0,1,1,0,1) = {4}:{6} w(2,0,1,1,0,1) = {3,4}:{9,10} w(0,1,1,1,0,1) = {5}:{6} w(1,1,1,1,0,1) = {2}:{9} w(2,1,1,1,0,1) = {1}:{3} w(0,2,1,1,0,1) = {10}:{11} w(1,2,1,1,0,1) = {1}:{7} w(2,2,1,1,0,1) = {2}:{3} w(0,0,2,1,0,1) = {9}:{11} w(1,0,2,1,0,1) = {4}:{5} w(2,0,2,1,0,1) = {3,4}:{7,8} w(0,1,2,1,0,1) = {5}:{6} w(1,1,2,1,0,1) = {2}:{10} w(2,1,2,1,0,1) = {1}:{3} w(0,2,2,1,0,1) = {9}:{11} w(1,2,2,1,0,1) = {1}:{8} w(2,2,2,1,0,1) = {2}:{3} w(0,0,0,2,0,1) = {10}:{11} w(1,0,0,2,0,1) = {4}:{5} w(2,0,0,2,0,1) = {9}:{11} w(0,1,0,2,0,1) = {7}:{11} w(1,1,0,2,0,1) = {6}:{8} w(2,1,0,2,0,1) = {1,4}:{3,10} w(0,2,0,2,0,1) = {2}:{11} w(1,2,0,2,0,1) = {6}:{10} w(2,2,0,2,0,1) = {2,4}:{3,8} w(0,0,1,2,0,1) = {10}:{11} w(1,0,1,2,0,1) = {1}:{3} w(2,0,1,2,0,1) = {4}:{11} w(0,1,1,2,0,1) = {9}:{11} w(1,1,1,2,0,1) = {2,4}:{3,5} w(2,1,1,2,0,1) = {9}:{11} w(0,2,1,2,0,1) = {9}:{11} w(1,2,1,2,0,1) = {1,4}:{3,5} w(2,2,1,2,0,1) = {7}:{11} w(0,0,2,2,0,1) = {2}:{11} w(1,0,2,2,0,1) = {2}:{3} w(2,0,2,2,0,1) = {4}:{11} w(0,1,2,2,0,1) = {10}:{11} w(1,1,2,2,0,1) = {2,4}:{3,6} w(2,1,2,2,0,1) = {10}:{11} w(0,2,2,2,0,1) = {10}:{11} w(1,2,2,2,0,1) = {1,4}:{3,6} w(2,2,2,2,0,1) = {8}:{11} w(0,0,0,0,1,1) = {}:{} w(1,0,0,0,1,1) = {8}:{11} w(2,0,0,0,1,1) = {8}:{11} w(0,1,0,0,1,1) = {1}:{7} w(1,1,0,0,1,1) = {2}:{8} w(2,1,0,0,1,1) = {2}:{9} w(0,2,0,0,1,1) = {2}:{11} w(1,2,0,0,1,1) = {1}:{10} w(2,2,0,0,1,1) = {1}:{7} w(0,0,1,0,1,1) = {2}:{3} w(1,0,1,0,1,1) = {5,9}:{7,11} w(2,0,1,0,1,1) = {2,11}:{4,10} w(0,1,1,0,1,1) = {2,3}:{9,11} w(1,1,1,0,1,1) = {3}:{8} w(2,1,1,0,1,1) = {9}:{7} w(0,2,1,0,1,1) = {2,3}:{10,11} w(1,2,1,0,1,1) = {3}:{10} w(2,2,1,0,1,1) = {7}:{9} w(0,0,2,0,1,1) = {2}:{3} w(1,0,2,0,1,1) = {6,8}:{10,11} w(2,0,2,0,1,1) = {1,11}:{4,7} w(0,1,2,0,1,1) = {2,3}:{10,11} w(1,1,2,0,1,1) = {3}:{7} w(2,1,2,0,1,1) = {10}:{8} w(0,2,2,0,1,1) = {2,3}:{9,11} w(1,2,2,0,1,1) = {3}:{9} w(2,2,2,0,1,1) = {8}:{10} w(0,0,0,1,1,1) = {9}:{10} w(1,0,0,1,1,1) = {3,4}:{9,11} w(2,0,0,1,1,1) = {2}:{11} w(0,1,0,1,1,1) = {4,5}:{10,11} w(1,1,0,1,1,1) = {6}:{11} w(2,1,0,1,1,1) = {2}:{10} w(0,2,0,1,1,1) = {1,4}:{9,11} w(1,2,0,1,1,1) = {6}:{11} w(2,2,0,1,1,1) = {1}:{8} w(0,0,1,1,1,1) = {1,4}:{10,11} w(1,0,1,1,1,1) = {4}:{6} w(2,0,1,1,1,1) = {7}:{11} w(0,1,1,1,1,1) = {5}:{6} w(1,1,1,1,1,1) = {2}:{9} w(2,1,1,1,1,1) = {1,3}:{4,9} w(0,2,1,1,1,1) = {2,3}:{10,11} w(1,2,1,1,1,1) = {1}:{7} w(2,2,1,1,1,1) = {2,3}:{4,7} w(0,0,2,1,1,1) = {6,9}:{7,10} w(1,0,2,1,1,1) = {4}:{5} w(2,0,2,1,1,1) = {10}:{11} w(0,1,2,1,1,1) = {5}:{6} w(1,1,2,1,1,1) = {2}:{10} w(2,1,2,1,1,1) = {1,3}:{4,10} w(0,2,2,1,1,1) = {2,3}:{9,11} w(1,2,2,1,1,1) = {1}:{8} w(2,2,2,1,1,1) = {2,3}:{4,8} w(0,0,0,2,1,1) = {10}:{11} w(1,0,0,2,1,1) = {2}:{4} w(2,0,0,2,1,1) = {9}:{11} w(0,1,0,2,1,1) = {4,6}:{9,11} w(1,1,0,2,1,1) = {7}:{8} w(2,1,0,2,1,1) = {1,2}:{3,11} w(0,2,0,2,1,1) = {1,4}:{10,11} w(1,2,0,2,1,1) = {9}:{10} w(2,2,0,2,1,1) = {1,2}:{3,11} w(0,0,1,2,1,1) = {1}:{7} w(1,0,1,2,1,1) = {4}:{6} w(2,0,1,2,1,1) = {4}:{6} w(0,1,1,2,1,1) = {2,3}:{9,11} w(1,1,1,2,1,1) = {3,6}:{4,9} w(2,1,1,2,1,1) = {5,10}:{8,11} w(0,2,1,2,1,1) = {9}:{11} w(1,2,1,2,1,1) = {3,6}:{4,7} w(2,2,1,2,1,1) = {5,8}:{10,11}

31 w(0,0,2,2,1,1) = {7}:{8} w(1,0,2,2,1,1) = {4}:{5} w(2,0,2,2,1,1) = {4}:{5} w(0,1,2,2,1,1) = {2,3}:{10,11} w(1,1,2,2,1,1) = {3,5}:{4,10} w(2,1,2,2,1,1) = {6,9}:{7,11} w(0,2,2,2,1,1) = {10}:{11} w(1,2,2,2,1,1) = {3,5}:{4,8} w(2,2,2,2,1,1) = {6,7}:{9,11} w(0,0,0,0,2,1) = {}:{} w(1,0,0,0,2,1) = {8}:{11} w(2,0,0,0,2,1) = {8}:{11} w(0,1,0,0,2,1) = {1}:{7} w(1,1,0,0,2,1) = {6}:{10} w(2,1,0,0,2,1) = {5}:{9} w(0,2,0,0,2,1) = {3}:{11} w(1,2,0,0,2,1) = {6}:{8} w(2,2,0,0,2,1) = {5}:{7} w(0,0,1,0,2,1) = {2}:{3} w(1,0,1,0,2,1) = {7}:{11} w(2,0,1,0,2,1) = {3}:{11} w(0,1,1,0,2,1) = {2,3}:{9,11} w(1,1,1,0,2,1) = {1}:{8} w(2,1,1,0,2,1) = {5,6}:{9,10} w(0,2,1,0,2,1) = {2,3}:{10,11} w(1,2,1,0,2,1) = {2}:{10} w(2,2,1,0,2,1) = {5,6}:{7,8} w(0,0,2,0,2,1) = {2}:{3} w(1,0,2,0,2,1) = {10}:{11} w(2,0,2,0,2,1) = {3}:{11} w(0,1,2,0,2,1) = {2,3}:{10,11} w(1,1,2,0,2,1) = {1}:{7} w(2,1,2,0,2,1) = {5,6}:{9,10} w(0,2,2,0,2,1) = {2,3}:{9,11} w(1,2,2,0,2,1) = {2}:{9} w(2,2,2,0,2,1) = {5,6}:{7,8} w(0,0,0,1,2,1) = {10}:{11} w(1,0,0,1,2,1) = {9}:{11} w(2,0,0,1,2,1) = {1,11}:{3,4} w(0,1,0,1,2,1) = {5}:{11} w(1,1,0,1,2,1) = {3}:{4} w(2,1,0,1,2,1) = {10}:{8} w(0,2,0,1,2,1) = {1}:{11} w(1,2,0,1,2,1) = {3}:{4} w(2,2,0,1,2,1) = {8}:{10} w(0,0,1,1,2,1) = {2}:{3} w(1,0,1,1,2,1) = {4}:{6} w(2,0,1,1,2,1) = {10}:{11} w(0,1,1,1,2,1) = {5}:{6} w(1,1,1,1,2,1) = {1}:{2} w(2,1,1,1,2,1) = {1}:{11} w(0,2,1,1,2,1) = {2,3}:{10,11} w(1,2,1,1,2,1) = {2}:{1} w(2,2,1,1,2,1) = {2}:{11} w(0,0,2,1,2,1) = {5,9}:{7,10} w(1,0,2,1,2,1) = {4}:{5} w(2,0,2,1,2,1) = {7}:{11} w(0,1,2,1,2,1) = {5}:{6} w(1,1,2,1,2,1) = {1}:{2} w(2,1,2,1,2,1) = {1}:{11} w(0,2,2,1,2,1) = {2,3}:{9,11} w(1,2,2,1,2,1) = {2}:{1} w(2,2,2,1,2,1) = {2}:{11} w(0,0,0,2,2,1) = {6}:{11} w(1,0,0,2,2,1) = {7}:{11} w(2,0,0,2,2,1) = {3,4}:{9,11} w(0,1,0,2,2,1) = {6}:{11} w(1,1,0,2,2,1) = {2,6}:{4,5} w(2,1,0,2,2,1) = {3,11}:{4,8} w(0,2,0,2,2,1) = {1}:{11} w(1,2,0,2,2,1) = {1,6}:{4,5} w(2,2,0,2,2,1) = {3,11}:{4,10} w(0,0,1,2,2,1) = {1}:{7} w(1,0,1,2,2,1) = {7}:{8} w(2,0,1,2,2,1) = {3,4}:{5,6} w(0,1,1,2,2,1) = {2,3}:{9,11} w(1,1,1,2,2,1) = {3,4}:{8,9} w(2,1,1,2,2,1) = {5,10}:{8,9} w(0,2,1,2,2,1) = {9}:{11} w(1,2,1,2,2,1) = {3,4}:{7,10} w(2,2,1,2,2,1) = {5,8}:{7,10} w(0,0,2,2,2,1) = {9}:{11} w(1,0,2,2,2,1) = {10}:{9} w(2,0,2,2,2,1) = {3,4}:{5,6} w(0,1,2,2,2,1) = {2,3}:{10,11} w(1,1,2,2,2,1) = {3,4}:{7,10} w(2,1,2,2,2,1) = {6,9}:{7,10} w(0,2,2,2,2,1) = {10}:{11} w(1,2,2,2,2,1) = {3,4}:{8,9} w(2,2,2,2,2,1) = {6,7}:{8,9} w(0,0,0,0,0,2) = {10}:{11} w(1,0,0,0,0,2) = {8}:{11} w(2,0,0,0,0,2) = {8}:{11} w(0,1,0,0,0,2) = {1}:{11} w(1,1,0,0,0,2) = {2}:{7} w(2,1,0,0,0,2) = {2}:{10} w(0,2,0,0,0,2) = {1}:{11} w(1,2,0,0,0,2) = {1}:{9} w(2,2,0,0,0,2) = {1}:{8} w(0,0,1,0,0,2) = {3}:{4} w(1,0,1,0,0,2) = {10}:{11} w(2,0,1,0,0,2) = {3}:{11} w(0,1,1,0,0,2) = {9}:{11} w(1,1,1,0,0,2) = {3,6}:{5,9} w(2,1,1,0,0,2) = {6}:{8} w(0,2,1,0,0,2) = {10}:{11} w(1,2,1,0,0,2) = {3,6}:{5,7} w(2,2,1,0,0,2) = {6}:{10} w(0,0,2,0,0,2) = {3}:{10} w(1,0,2,0,0,2) = {7}:{11} w(2,0,2,0,0,2) = {3}:{11} w(0,1,2,0,0,2) = {10}:{11} w(1,1,2,0,0,2) = {3,5}:{6,10} w(2,1,2,0,0,2) = {5}:{7} w(0,2,2,0,0,2) = {9}:{11} w(1,2,2,0,0,2) = {3,5}:{6,8} w(2,2,2,0,0,2) = {5}:{9} w(0,0,0,1,0,2) = {10}:{11} w(1,0,0,1,0,2) = {9}:{11} w(2,0,0,1,0,2) = {4}:{5} w(0,1,0,1,0,2) = {2}:{11} w(1,1,0,1,0,2) = {2,4}:{3,8} w(2,1,0,1,0,2) = {6}:{10} w(0,2,0,1,0,2) = {7}:{11} w(1,2,0,1,0,2) = {1,4}:{3,10} w(2,2,0,1,0,2) = {6}:{8} w(0,0,1,1,0,2) = {2}:{11} w(1,0,1,1,0,2) = {4}:{11} w(2,0,1,1,0,2) = {2}:{3} w(0,1,1,1,0,2) = {10}:{11} w(1,1,1,1,0,2) = {8}:{11} w(2,1,1,1,0,2) = {1,4}:{3,6}

32 w(0,2,1,1,0,2) = {10}:{11} w(1,2,1,1,0,2) = {10}:{11} w(2,2,1,1,0,2) = {2,4}:{3,6} w(0,0,2,1,0,2) = {10}:{11} w(1,0,2,1,0,2) = {4}:{11} w(2,0,2,1,0,2) = {1}:{3} w(0,1,2,1,0,2) = {9}:{11} w(1,1,2,1,0,2) = {7}:{11} w(2,1,2,1,0,2) = {1,4}:{3,5} w(0,2,2,1,0,2) = {9}:{11} w(1,2,2,1,0,2) = {9}:{11} w(2,2,2,1,0,2) = {2,4}:{3,5} w(0,0,0,2,0,2) = {10}:{11} w(1,0,0,2,0,2) = {10}:{11} w(2,0,0,2,0,2) = {9}:{11} w(0,1,0,2,0,2) = {2}:{11} w(1,1,0,2,0,2) = {2}:{11} w(2,1,0,2,0,2) = {10}:{8} w(0,2,0,2,0,2) = {8}:{11} w(1,2,0,2,0,2) = {1}:{11} w(2,2,0,2,0,2) = {8}:{10} w(0,0,1,2,0,2) = {9}:{11} w(1,0,1,2,0,2) = {3,4}:{7,8} w(2,0,1,2,0,2) = {4}:{5} w(0,1,1,2,0,2) = {9}:{11} w(1,1,1,2,0,2) = {2}:{3} w(2,1,1,2,0,2) = {1}:{8} w(0,2,1,2,0,2) = {5}:{6} w(1,2,1,2,0,2) = {1}:{3} w(2,2,1,2,0,2) = {2}:{10} w(0,0,2,2,0,2) = {2,10}:{3,11} w(1,0,2,2,0,2) = {3,4}:{9,10} w(2,0,2,2,0,2) = {4}:{6} w(0,1,2,2,0,2) = {10}:{11} w(1,1,2,2,0,2) = {2}:{3} w(2,1,2,2,0,2) = {1}:{7} w(0,2,2,2,0,2) = {5}:{6} w(1,2,2,2,0,2) = {1}:{3} w(2,2,2,2,0,2) = {2}:{9} w(0,0,0,0,1,2) = {}:{} w(1,0,0,0,1,2) = {8}:{11} w(2,0,0,0,1,2) = {8}:{11} w(0,1,0,0,1,2) = {3}:{11} w(1,1,0,0,1,2) = {5}:{7} w(2,1,0,0,1,2) = {6}:{8} w(0,2,0,0,1,2) = {1}:{7} w(1,2,0,0,1,2) = {5}:{9} w(2,2,0,0,1,2) = {6}:{10} w(0,0,1,0,1,2) = {2}:{3} w(1,0,1,0,1,2) = {3}:{11} w(2,0,1,0,1,2) = {10}:{11} w(0,1,1,0,1,2) = {2,3}:{9,11} w(1,1,1,0,1,2) = {5,6}:{7,8} w(2,1,1,0,1,2) = {2}:{9} w(0,2,1,0,1,2) = {2,3}:{10,11} w(1,2,1,0,1,2) = {5,6}:{9,10} w(2,2,1,0,1,2) = {1}:{7} w(0,0,2,0,1,2) = {2}:{3} w(1,0,2,0,1,2) = {3}:{11} w(2,0,2,0,1,2) = {7}:{11} w(0,1,2,0,1,2) = {2,3}:{10,11} w(1,1,2,0,1,2) = {5,6}:{7,8} w(2,1,2,0,1,2) = {2}:{10} w(0,2,2,0,1,2) = {2,3}:{9,11} w(1,2,2,0,1,2) = {5,6}:{9,10} w(2,2,2,0,1,2) = {1}:{8} w(0,0,0,1,1,2) = {6}:{11} w(1,0,0,1,1,2) = {3,4}:{9,11} w(2,0,0,1,1,2) = {7}:{11} w(0,1,0,1,1,2) = {1}:{11} w(1,1,0,1,1,2) = {3,11}:{4,10} w(2,1,0,1,1,2) = {1,6}:{4,5} w(0,2,0,1,1,2) = {6}:{11} w(1,2,0,1,1,2) = {3,11}:{4,8} w(2,2,0,1,1,2) = {2,6}:{4,5} w(0,0,1,1,1,2) = {9}:{11} w(1,0,1,1,1,2) = {3,4}:{5,6} w(2,0,1,1,1,2) = {10}:{9} w(0,1,1,1,1,2) = {10}:{11} w(1,1,1,1,1,2) = {6,7}:{8,9} w(2,1,1,1,1,2) = {3,4}:{8,9} w(0,2,1,1,1,2) = {2,3}:{10,11} w(1,2,1,1,1,2) = {6,9}:{7,10} w(2,2,1,1,1,2) = {3,4}:{7,10} w(0,0,2,1,1,2) = {1}:{7} w(1,0,2,1,1,2) = {3,4}:{5,6} w(2,0,2,1,1,2) = {7}:{8} w(0,1,2,1,1,2) = {9}:{11} w(1,1,2,1,1,2) = {5,8}:{7,10} w(2,1,2,1,1,2) = {3,4}:{7,10} w(0,2,2,1,1,2) = {2,3}:{9,11} w(1,2,2,1,1,2) = {5,10}:{8,9} w(2,2,2,1,1,2) = {3,4}:{8,9} w(0,0,0,2,1,2) = {10}:{11} w(1,0,0,2,1,2) = {1,11}:{3,4} w(2,0,0,2,1,2) = {9}:{11} w(0,1,0,2,1,2) = {1}:{11} w(1,1,0,2,1,2) = {8}:{10} w(2,1,0,2,1,2) = {3}:{4} w(0,2,0,2,1,2) = {5}:{11} w(1,2,0,2,1,2) = {10}:{8} w(2,2,0,2,1,2) = {3}:{4} w(0,0,1,2,1,2) = {5,9}:{7,10} w(1,0,1,2,1,2) = {7}:{11} w(2,0,1,2,1,2) = {4}:{5} w(0,1,1,2,1,2) = {2,3}:{9,11} w(1,1,1,2,1,2) = {2}:{11} w(2,1,1,2,1,2) = {2}:{1} w(0,2,1,2,1,2) = {5}:{6} w(1,2,1,2,1,2) = {1}:{11} w(2,2,1,2,1,2) = {1}:{2} w(0,0,2,2,1,2) = {2}:{3} w(1,0,2,2,1,2) = {10}:{11} w(2,0,2,2,1,2) = {4}:{6} w(0,1,2,2,1,2) = {2,3}:{10,11} w(1,1,2,2,1,2) = {2}:{11} w(2,1,2,2,1,2) = {2}:{1} w(0,2,2,2,1,2) = {5}:{6} w(1,2,2,2,1,2) = {1}:{11} w(2,2,2,2,1,2) = {1}:{2} w(0,0,0,0,2,2) = {}:{} w(1,0,0,0,2,2) = {8}:{11} w(2,0,0,0,2,2) = {8}:{11} w(0,1,0,0,2,2) = {2}:{11} w(1,1,0,0,2,2) = {1}:{7} w(2,1,0,0,2,2) = {1}:{10} w(0,2,0,0,2,2) = {1}:{7} w(1,2,0,0,2,2) = {2}:{9} w(2,2,0,0,2,2) = {2}:{8} w(0,0,1,0,2,2) = {2}:{3} w(1,0,1,0,2,2) = {1,11}:{4,7} w(2,0,1,0,2,2) = {6,8}:{10,11}

33 w(0,1,1,0,2,2) = {2,3}:{9,11} w(1,1,1,0,2,2) = {8}:{10} w(2,1,1,0,2,2) = {3}:{9} w(0,2,1,0,2,2) = {2,3}:{10,11} w(1,2,1,0,2,2) = {10}:{8} w(2,2,1,0,2,2) = {3}:{7} w(0,0,2,0,2,2) = {2}:{3} w(1,0,2,0,2,2) = {2,11}:{4,10} w(2,0,2,0,2,2) = {5,9}:{7,11} w(0,1,2,0,2,2) = {2,3}:{10,11} w(1,1,2,0,2,2) = {7}:{9} w(2,1,2,0,2,2) = {3}:{10} w(0,2,2,0,2,2) = {2,3}:{9,11} w(1,2,2,0,2,2) = {9}:{7} w(2,2,2,0,2,2) = {3}:{8} w(0,0,0,1,2,2) = {10}:{11} w(1,0,0,1,2,2) = {9}:{11} w(2,0,0,1,2,2) = {2}:{4} w(0,1,0,1,2,2) = {1,4}:{10,11} w(1,1,0,1,2,2) = {1,2}:{3,11} w(2,1,0,1,2,2) = {9}:{10} w(0,2,0,1,2,2) = {4,6}:{9,11} w(1,2,0,1,2,2) = {1,2}:{3,11} w(2,2,0,1,2,2) = {7}:{8} w(0,0,1,1,2,2) = {7}:{8} w(1,0,1,1,2,2) = {4}:{5} w(2,0,1,1,2,2) = {4}:{5} w(0,1,1,1,2,2) = {10}:{11} w(1,1,1,1,2,2) = {6,7}:{9,11} w(2,1,1,1,2,2) = {3,5}:{4,8} w(0,2,1,1,2,2) = {2,3}:{10,11} w(1,2,1,1,2,2) = {6,9}:{7,11} w(2,2,1,1,2,2) = {3,5}:{4,10} w(0,0,2,1,2,2) = {1}:{7} w(1,0,2,1,2,2) = {4}:{6} w(2,0,2,1,2,2) = {4}:{6} w(0,1,2,1,2,2) = {9}:{11} w(1,1,2,1,2,2) = {5,8}:{10,11} w(2,1,2,1,2,2) = {3,6}:{4,7} w(0,2,2,1,2,2) = {2,3}:{9,11} w(1,2,2,1,2,2) = {5,10}:{8,11} w(2,2,2,1,2,2) = {3,6}:{4,9} w(0,0,0,2,2,2) = {9}:{10} w(1,0,0,2,2,2) = {2}:{11} w(2,0,0,2,2,2) = {3,4}:{9,11} w(0,1,0,2,2,2) = {1,4}:{9,11} w(1,1,0,2,2,2) = {1}:{8} w(2,1,0,2,2,2) = {6}:{11} w(0,2,0,2,2,2) = {4,5}:{10,11} w(1,2,0,2,2,2) = {2}:{10} w(2,2,0,2,2,2) = {6}:{11} w(0,0,1,2,2,2) = {6,9}:{7,10} w(1,0,1,2,2,2) = {10}:{11} w(2,0,1,2,2,2) = {4}:{5} w(0,1,1,2,2,2) = {2,3}:{9,11} w(1,1,1,2,2,2) = {2,3}:{4,8} w(2,1,1,2,2,2) = {1}:{8} w(0,2,1,2,2,2) = {5}:{6} w(1,2,1,2,2,2) = {1,3}:{4,10} w(2,2,1,2,2,2) = {2}:{10} w(0,0,2,2,2,2) = {1,4}:{10,11} w(1,0,2,2,2,2) = {7}:{11} w(2,0,2,2,2,2) = {4}:{6} w(0,1,2,2,2,2) = {2,3}:{10,11} w(1,1,2,2,2,2) = {2,3}:{4,7} w(2,1,2,2,2,2) = {1}:{7} w(0,2,2,2,2,2) = {5}:{6} w(1,2,2,2,2,2) = {1,3}:{4,9} w(2,2,2,2,2,2) = {2}:{9} List 1 1:= <, 0:= =, 2:= >, and w(x) is the weighing in the direction x.

34 The completed map of the first algorithm to sort 11 coins f(0,0,0,0,0,0) = 2047 f(1,0,0,0,0,0,0) = 1468 f(2,0,0,0,0,0,0) = 579 f(0,1,0,0,0,0,0) = 1855 f(1,1,0,0,0,0,0) = 2046 f(2,1,0,0,0,0,0) = 2 f(0,2,0,0,0,0,0) = 192 f(1,2,0,0,0,0,0) = 2045 f(2,2,0,0,0,0,0) = 1 f(0,0,1,0,0,0,0) = 138 f(1,0,1,0,0,0,0) = 1786 f(2,0,1,0,0,0,0) = 134 f(0,1,1,0,0,0,0) = 1974 f(1,1,1,0,0,0,0) = 944 f(2,1,1,0,0,0,0) = 1551 f(0,2,1,0,0,0,0) = 137 f(1,2,1,0,0,0,0) = 752 f(2,2,1,0,0,0,0) = 1167 f(0,0,2,0,0,0,0) = 1909 f(1,0,2,0,0,0,0) = 1913 f(2,0,2,0,0,0,0) = 261 f(0,1,2,0,0,0,0) = 1910 f(1,1,2,0,0,0,0) = 880 f(2,1,2,0,0,0,0) = 1295 f(0,2,2,0,0,0,0) = 73 f(1,2,2,0,0,0,0) = 496 f(2,2,2,0,0,0,0) = 1103 f(1,0,0,1,0,0,0) = 2044 f(2,0,0,1,0,0,0) = 1991 f(0,1,0,1,0,0,0) = 1838 f(1,1,0,1,0,0,0) = 300 f(2,1,0,1,0,0,0) = 774 f(0,2,0,1,0,0,0) = 225 f(1,2,0,1,0,0,0) = 108 f(2,2,0,1,0,0,0) = 197 f(1,0,1,1,0,0,0) = 696 f(2,0,1,1,0,0,0) = 2031 f(0,1,1,1,0,0,0) = 1674 f(1,1,1,1,0,0,0) = 1594 f(2,1,1,1,0,0,0) = 1827 f(0,2,1,1,0,0,0) = 128 f(1,2,1,1,0,0,0) = 1209 f(2,2,1,1,0,0,0) = 1251 f(1,0,2,1,0,0,0) = 376 f(2,0,2,1,0,0,0) = 2015 f(0,1,2,1,0,0,0) = 1354 f(1,1,2,1,0,0,0) = 1338 f(2,1,2,1,0,0,0) = 1811 f(0,2,2,1,0,0,0) = 64 f(1,2,2,1,0,0,0) = 1145 f(2,2,2,1,0,0,0) = 1235 f(1,0,0,2,0,0,0) = 56 f(2,0,0,2,0,0,0) = 3 f(0,1,0,2,0,0,0) = 1822 f(1,1,0,2,0,0,0) = 1850 f(2,1,0,2,0,0,0) = 1939 f(0,2,0,2,0,0,0) = 209 f(1,2,0,2,0,0,0) = 1273 f(2,2,0,2,0,0,0) = 1747 f(1,0,1,2,0,0,0) = 32 f(2,0,1,2,0,0,0) = 1671 f(0,1,1,2,0,0,0) = 1983 f(1,1,1,2,0,0,0) = 812 f(2,1,1,2,0,0,0) = 902 f(0,2,1,2,0,0,0) = 693 f(1,2,1,2,0,0,0) = 236 f(2,2,1,2,0,0,0) = 709 f(1,0,2,2,0,0,0) = 16 f(2,0,2,2,0,0,0) = 1351 f(0,1,2,2,0,0,0) = 1919 f(1,1,2,2,0,0,0) = 796 f(2,1,2,2,0,0,0) = 838 f(0,2,2,2,0,0,0) = 373 f(1,2,2,2,0,0,0) = 220 f(2,2,2,2,0,0,0) = 453 f(0,0,0,0,1,0,0) = 1984 f(1,0,0,0,1,0,0) = 1534 f(2,0,0,0,1,0,0) = 66 f(0,1,0,0,1,0,0) = 1804 f(1,1,0,0,1,0,0) = 1854 f(2,1,0,0,1,0,0) = 1943 f(0,2,0,0,1,0,0) = 245 f(1,2,0,0,1,0,0) = 1277 f(2,2,0,0,1,0,0) = 1751 f(0,0,1,0,1,0,0) = 1060 f(2,0,1,0,1,0,0) = 710 f(1,1,1,0,1,0,0) = 1832 f(2,1,1,0,1,0,0) = 1702 f(1,2,1,0,1,0,0) = 1256 f(2,2,1,0,1,0,0) = 1701 f(0,0,2,0,1,0,0) = 1003 f(2,0,2,0,1,0,0) = 837 f(1,1,2,0,1,0,0) = 1816 f(2,1,2,0,1,0,0) = 1366 f(1,2,2,0,1,0,0) = 1240 f(2,2,2,0,1,0,0) = 1365 f(0,0,0,1,1,0,0) = 1134 f(2,0,0,1,1,0,0) = 388 f(1,1,0,1,1,0,0) = 1662 f(2,1,0,1,1,0,0) = 919 f(1,2,0,1,1,0,0) = 1469 f(2,2,0,1,1,0,0) = 727 f(0,0,1,1,1,0,0) = 2020 f(1,0,1,1,1,0,0) = 648 f(2,0,1,1,1,0,0) = 1699 f(0,1,1,1,1,0,0) = 1930 f(1,1,1,1,1,0,0) = 1714 f(2,1,1,1,1,0,0) = 1538 f(1,2,1,1,1,0,0) = 1713 f(2,2,1,1,1,0,0) = 1153 f(0,0,2,1,1,0,0) = 2027 f(1,0,2,1,1,0,0) = 328 f(2,0,2,1,1,0,0) = 1363 f(0,1,2,1,1,0,0) = 1866 f(1,1,2,1,1,0,0) = 1394 f(2,1,2,1,1,0,0) = 1282 f(1,2,2,1,1,0,0) = 1393 f(2,2,2,1,1,0,0) = 1089 f(1,0,0,2,1,0,0) = 392 f(1,1,0,2,1,0,0) = 2034 f(2,1,0,2,1,0,0) = 1799 f(1,2,0,2,1,0,0) = 2033 f(2,2,0,2,1,0,0) = 1223