PERFECT MAGIC CUBES OF ORDER 4 m

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1 PERFECT MAGIC CUBES OF ORDER BRIAN ALSPACH and KATHERINE HEINRCH Sion Fraser University, Burnaby, B. C., Canada VA S ABSTRACT It has long been known that there exists a perfect agic cube of order n where n, 9, and k with odd and >_. That they do not exist for orders,, and is not difficult to show. Recently, several authors have constructed perfect agic cubes of order. We shall give a ethod for constructing perfect agic cubes of orders n - k with odd and _>.?. INTRODUCTION A agic square of order n is an nx n arrangeent of the integers,,.., n so that the su of the integers in every row, colun and the two ain diagonals is n(n + )% the agic su. Magic squares of orders and are shown in Figure It is a well-known and long established fact that there exists a agic square of every order n, n ^. For details of these constructions, the reader is referred to W. S. Andrews [], Maurice Kraitchik [], and W. W. Rouse Ball []. We can extend the concept of agic squares into three diensions. A agic cube of order n is an nxnxn arrangeent of the integers,,..., n so that the su of the integers in every row, colun, file and space diagonal (of which there are four) is n(n + I)/i the agic su. A agic cube of order is exhibited in Figure Fig. Magic cubes can be constructed for every order n, n ^ (see W. S. Andrews []). A perfect agpie cube of order n is a agic cube of order n with the additional property that the su of the integers in the ain diagonals of every layer parallel to a face of the cube is also n(n + l)/. In 99 Barkley Rosser and R. J. Walker [] showed that there exists a perfect agic cube of order n, n ^,,,, or, odd. In fact, they constructed diabolic agic cubes of order n and showed that they exist only when n,,,, or, odd. A diabolic (or pandiagonal) agic cube of order n is a agic cube of order n in which the su of the integers in every diagonal, both broken and unbroken, is n(n + l)/. Clearly, a diabolic agic cube is also a perfect agic cube. We shall prove that there do not exist perfect agic cubes of orders and. These proofs are due to Lewis Myers, Jr. [9] and Richard Schroeppel [9]. Perfect agic cubes of order are known to have been constructed by lanp, Howard, Richard Schroeppel, Ernst G. Straus, and Bayard E. Wynne. In this paper, we 9

2 98 PERFECT MAGIC CUBES OF ORDER k [April shall present a construction for perfect agic cubes of orders n k 9 odd, _>, leaving only the orders n,, 0, and for odd to be resolved. We reark here that Schroeppel has shown that if a perfect agic cube of order exists, then its center ust be.. VETWT0NS avid CONSTRUCTIONS As far as possible, the definitions will be in accord with those given in J. Denes and A. D. Keedwell []. An nxnxn three-diensional atrix coprising n files, each having n rows and n coluns, is called a cubic array of order n. We shall write this array as A (a^-fe), i, j, k e {,,..., n} where dij k is the eleent in the ith row, jth colun, and kth. file of the array. When we write a^ + % j + s, k + t we ean for the indices i + r 9 j + s, and k + t to be calculated odulo n on the residues J-, ^ 9.. a, fl a The set of n eleents {cii + i, j t k> &,,..., n} constitutes a colun; {ai^ + ix*,,..., n} constitutes a row; {&,j k + si : & 9,..., n} constitutes a file; and {a^ + t j + > fe. &,,..., n}, {a i + l^^ k + Si i I,,..., n}, {^i, J - +, k + : & l',,'..., n} and {a i +, J. + s fc + : *',,..., n] constitute the diagonals. Note that a diagonal is either broken or unbroken; being unbroken if all n of its eleents lie on a straight line. The unbroken diagonals consist of the ain diagonals, of which there are two in every layer parallel to a face of the cube, and the four space diagonals. We shall distinguish three types of layers in a cube. There are those with fixed row, fixed colun, or fixed file. The first we shall call the CF-layers, the ith. CF-layer consisting of the n eleents {a^ki j', k <_ n}. The secone are the RF-layers in which the jth RF-layer consists of the n eleents {a j k : <. i 9 k <_ n}. And finally, the RC-layers, the kth consisting of the eleents {a^k. <_ i 9 j <_ n}. A cubic array of order n is called a Latin cube of order n if it has n distinct eleents each repeated n ties and so arranged that in each layer parallel to a face of the cube all n distinct eleents appear, and each is repeated exactly n ties in that layer. In the case when each layer parallel to.a face of the cube is a Latin square, we have what is called a perutation cube of order n. Fro this point on, we shall be concerned only with Latin cubes (and perutation cubes) based on the integers,,..., n* Three Latin cubes of order n, A {a^k) 9 B (bij k ) ai 9 *d C {o^k) 9 are said to be orthogonal if aong the n ordered -tuples of eleents (a i{j - k9 b^k9 e ijk) every distinct ordered -tuple involving the integers,,...,n occurs exactly once. Should A 9 B S and C be orthogonal perutation cubes, they are said to for a variational cube. We shall write D (d i;] - k ) where d^k - (a^- k9 b^k9 ijk) t o ^ e t n e cube obtaineid on superiposing the Latin cubes A 9 B 9 and 'C and will denote it by D 0, B 9 C). A cubic array of order n in which each of the integers,,..., n occurs exactly once and in which the su of the integers in every row, colun, file, and unbroken diagonal is n{n + l)/ is called a perfect agic cube. We shall give two ethods for constructing perfect agic cubes. These ethods for the basis on which the perfect agic cubes of order n- h 9 odd,? J>, of Section will be constructed. CotidtlOYl?** Let A (a^-fc), B (b ijk ) 9 and C (c i;jk ) be three orthogonal Latin cubes of order n with the property that in each cube the su of the integers in every row, colun, file, and unbroken diagonal is n(n+ l)/. Then the cube E (e ijk ) where e ijk n (a ijk - ) + n(b ijk - ) + (c ijk - ) + is a perfect agic cube of order n. This is verified by checking that each of the integers,,..., n appears in E and that the su of the integers in every row,

3 98] PERFECT MAGIC CUBES OF ORDER k 99 colun, file, and unbroken diagonal i s n ( n + l ) /. It is clear that each of 9 9.., n appears exactly once in E. We shall show that the su of the integers in any row of E is n(n + l)/. The reaining sus can be checked in a siilar anner. n n n n n n ~ n Y, a i.j+i.k +n Y, b *.d+*>*- + L^i^.fc - E (n + n ) (n + n + l)(n(n + l)/) -n(n + n) n(n + l)/. CoviA&LULCLtLon : Let A (a^k) and fciok) be perfect agic cubes of orders tfz and n, respectively. Replace b^k in by the cube C - (c rst ) where e r8t - + CL vs ~ ) This results in a perfect agic cube E - (e ^- k ) of order ra?. Each of the integers,,..., IT! +, H +,..., T, (n - l) +, (n - l V +-,..., n appears exactly once in E\ As in the first construction, we shall show that the row su in E is n((n) + l)/; the reaining sus are siilarly verified. n_ n SL I wn(rn + l)/ + w (n (n + l)/ -- n) n((n) + l)/. It will be seen in Theore. that it is not necessary that A and B should both be perfect agic cubes in order for E to be a perfect agic cube.. VERFECT MAGIC CUBES The first result is stated without proof and is due to Barkley Rosser and R. J. Walker []. ThzoKo.: There exists a perfect agic cube of order n provided n,,,, or k for odd.a The following three theores are the only known nonexistence results for perfect agic cubes. For the first, the proof is trivial. The proof of the second theore is that of Lewis Myers, Jr. (see [9]) and of the third is that of Richard Schroeppel (also see [9]). Tk&QJie.: There is no perfect agic cube of order.a ThdOKo.: There is no perfect agic cube of order. V*WOfa Let A (a^k) be a perfect agic cube of order ; the agic su is. The following equations ust all hold: a llk + a k + a k a lk + a zzk + a ik a lk + a k + a k ^ and a llk + a lzk + a k a k + a zk + a k. But together these iply that a, for k,, and, a contradiction.

4 00 PERFECT MAGIC CUBES OF ORDER k [April Tk&OKQjn,: There is no perfect agic cube of order. VK.OO^: Let A ( a ^ ) be a perfect agic cube of order ; the agic su is 0. First, we shall show that in any layer of such a cube the su of the four corner eleents is 0. Consider the kth RC-layer. The following equations ust hold in A: a llk + a k + a k + a lhk a llk + a k + a fc + a hhk a llk + a k + a k + a hlk 0, a lhk + a Zk + a k + a hlk a Zffc + a hk + a htk + a hhtk These iply that fl flk + ^ f c + ^fe + a^k 0 - and as then fan* + a ife + a ifc + a^fe> + E jl a Hk E X > ^ - 0 > ' 0 a ilk + *H>* + a fe + * W 0 - Since the sae arguent holds for any type of layer in the cube, we have that the su of the four corner eleents in any layer is 0. A siilar arguent shows that cz + a ( + a hhl + a hhh 0. Thus we have <*iiv+ a iif + "i* + a i. i a i**i + a n + a ^ h + % fro which it follows that ^ ^ - ^if Xf tf ' ^. " J - J ^ ' j a ill + a HH + ^ + a!l + K * + Sin) 0 ' and hence a ^ ^ + a hhl, Siilarly, we can show that a lhl + a hhl. Cobining these two results, we have a lhl - a hhh9 a contradiction. Using an arguent siilar to that of Theore. Schroeppel has shown that if there exists a perfect agic cube of order its center is. For soe tie it was not generally known whether or not there existed a perfect agic cube of order but when, in 9, Martin Gardner [9] asked for such a cube, it appeared that they had been constructed without difficulty by any authors including Schroeppel, Ian P. Howard, Ernst G. Straus, and Bayard E. Wynne []. Tk<L0h.,: There exists a perfect agic cube of order. VK.00^: We shall construct a variational cube of order fro which a perfect agic cube of order can be obtained via Construction. Let the three cubes foring the variational cube be A 9 B 9 and C; the first RC-layer of each being shown in Figure. Coplete A 9 B 9 and C using the defining relations a i, j, fc + i a iok + L > b t,j,k + l b idk + a n d %d.k + l C M + > where the addition is odulo on the residues,,...,. Now, in A and B 9 exchange the integers and throughout each cube. The variational cube of order now has the properties required by Construction and so we can construct a perfect agic cube of order. This can easily be checked.

5 98] PERFECT MAGIC CUBES OF O R D E R S 0 ' Fig. We shall now proceed to the ain theore. T/ieo-fce.: There exists a perfect agic cube of. order k for odd and _>. VK.OO^I We know that there exists a perfect agic cube of order, odd and _>. This follows fro Theores. and,. Since there does not exist a perfect agic cube of order (Theore.), we cannot siply appeal to Construction and obtain the desired perfect agic cubes. However, we can use Construction and by a suitable arrangeent of cubes of order obtain a perfect agic cube of order. The construction is as follows. Let A ( a ^ ) be a perfect agic cube of order 9 odd and J>. Let (fc^k) be a cubic array of order in which each fc^-fc is soe cubic array Z^k of order whose entries are ordered -tuples fro the integers,,, with every such -tuple appearing exactly once. The D^^ are to be chosen in such a way that in the cubic array B the coponentwise su of the integers in every row, colun, file, and unbroken diagonal is (0, 0???, 0). It is now a siple atter to produce a perfect agic cube of order. In D^k replace the -tuple (r, s, t) by the integer ((P - ) + (s - ) + (t - ) + ) + ( a ^ - ). The cubic array E (e^-fc) of order so constructed is, by considering Constructions and, a perfect agic cube. It reains then to deterine the order cubic arrays # # &. Consider the four Latin cubes X l9 X 9 X, and X^ as shown in Figure where fro left to right we have the first to the fourth RC-layers. It is not difficult to check that X, J, and X are orthogonal, as are I l X l9 and X. We shall write X^ for the Latin cube X± in which the integers and have been exchanged as have and. Also (X 9 X, Z ) f eans that the cubic array (X l9 X l9 X ) has been rotated forward through 90 so that RC-layers have becoe CFlayers, CF-layers have becoe RC-layers, and the roles of rows and files have interchanged in RF-layers. x l l X i X t - (continued)

6 0 PERFECT MAGIC CUBES OF ORDER k [April J i t : Fig. We can now define the cubic arrays B i. k. V + l + l + l + l * ' ' ^,, ^,, ^,, & i t l, D U 9 t ^ ' ^ ' ^,, ^ ' X * ^ ) X ) '» %,.,,.., - ^,, - ^, -, - ^ ' ^ ' ^ P ^,, ^, -, ( X > X h> X 0 ", a, - J ^, -, - V^» ^ ' ^ ^ ^,, ^, -, ( ^» ^» ^ ) Di, + - i, i ~ D i 9 -+l-i, + l - i ^. i. + l - i i,i t i t ^ l ' ^» ^ )» t,, + U i, + l - i, i ^ i t + l-i i + l - i "i t i 9 + l - i "i t i, i ^ ' ^» ^ ^» + +. In every CF-layer of B, except for the second and third, replace the reaining b^'ji in each unbroken diagonal by either *V* - <*» X > J > r D i ^ <*?» J?> J t) so that in each diagonal there are (? - l)/ arrays (X 9 X 9 X ) and ( - l)/ arrays (X\ 9 X*, X^). In the second and third layers do the sae but here there are to be only ( -. ) / of each type of array as already three arrays in each diagonal are deterined. All reaining b^k are to be replaced by D ijk ( X l» X > X ^ ' We ust now (verify that in this cubic array the coponentwise su of the integers in every row 9 colun, file, and unbroken diagonal is (IO, 0, IO). Since the su of the integers in every row, colun, and file of X^ and J*,,,,, is 0, then in B the coponentwise su of the integers in every row, colun, and file is (IO, IOTT?, 0). Also, as the su of the integers in every unbroken diagonal in the RC-layers and RF-layers of X^ and X^, i -,,,, is 0, and as (X 9 X 9 X ) f does not occur on any of these unbroken diagonals in B 9 then the coponentwise su of the integers in these unbroken diagonals of B is (IO, IO?, IO). So we now have only to check the sus on the unbroken diagonals of the CF-layers and the sus on the four space diagonals of B. The unbroken diagonals in the CF-layers of B are D^ll9 B il9» ^i anc * D ii> D i,-i t z> > D ii» ^ l >,...,. Let us write S r (D^k ) for the coponentwise su of the integers in the relevant diagonal in the pth CF-layer of ^ijk ^ e want to show that ] [ X (Z?^.) ^S r (D it +.. d t d ) (0, 0, 0), V,,,. j- J - l

7 98] PERFECT MAGIC CUBES OF ORDER k 0 If i + S, then X ( 0 W ) EL^±s r ax, x, x )) +!"-i-ls r «x* 9 x* 9 x*))+s P «x 9 x 9 x s y) j - i???? _ - - (8,, ) + ( 9, 8) + (0, 0, 0) when r (,, 8) + ~ (8,, ) + (0, 0, 0) when v (,, ) + ~ (8,, 8) + (0, 0, 0) when r i (8,, 8) + ~ (,, ) + (0, 0, 0) when r Also, (0, 0, 0). X X ( ^ ) J - l L ^ ^ ((*!> J, *» + ^ ^ ( ( *? 9 *?* X* ))+S P «X 9 X,, X x )) + r ((Z*, J, J*)) +, ( ( Z l f X 9 J ) f ) ^--^-(8,, ) + ^-^-(,, 8) + (, 8, 8) + (,, ) + (0, 0, 0) when r _ - (,, 8) + ^ ~ ^ ( 8 9, ) + (,, ) + (, 8, 8) + (0, 0, 0) when r - y l ( i,, ) + L _ ( 8,, 8) + (, 8, ) + (,, 8) + (0, 0, 0) when r ^ - y - ^ ( 8,, 8) +L^_(i,, ) + (,, 8) + (, 8, ) + (0, 0, 0) when r and (IO, IO, l O ) - r ^ ( U $ x 9 x )) + ~ - ^, ( ( z t *?. x*))'+.(u, x x )) + s (a*, x* xp) +s > (a, *, x )o ~^-(S 9, ) +^L-^(,, 8) + (, 8, ) + (8,, ( ) + (0, 0, 0) when r - y ^ (,, 8) + L _ ( 8,, ) + (8,, ) + (, 8, ) + (0, 0, 0) when r - ^ (,, ) + L _ ( 8,, 8) '+ + (,, ) + (8, 8, ) + (0, 0, 0) when r ^ - - ^ ( 8,, 8) + ^ - (,, ) + (8, 8, ) + (,, ) (0, 0, 0/), + (0, 0,. 0) when r

8 0 PERFECT MAGIC CUBES OF ORDER k [April Siilarly, one can check that H S r(vi, + l-j,j) (0W, 0/, 0/) and so the coponentwise su of the integers in the unbroken diagonals in the CF-layers of B is (0/, 0/, 0/). The four space diagonals of B axe Diii> l X» '» ' ^. + i-i,*' i.,..., TTZ; ^w+l-,t,» "^ ^» > '» > ^ + l-i,+l-i, i* ^»,...,. Write SiDtjk) to be the su of the integers in the relevant space diagonal of D J j,. We want to show that t * il (0, 0, 0/). Consider each of the space diagonals in turn. Y,s<P) sax, x, x )) + sax, x k, x x )) + sax, x lt x z )) + < n -^-s«x, x, x )) +? L^-s«x*, x*, x* )) (, 0, ) + (0,, ) + (,, 0) + ~ (, 0, ) (0, 0, 0) + - ^ (, 0, ) Y, s (Pi, + i-i,i) - S({X lt X, X )) + S«X, X, Z x )) + ( ( Z, X lt X )) + - Z _ S f ( ( j, j > j ) ) ^. ^ ^ - ^ ( ( j *, j *, j * ) ) (, 0, ) + (0,, ) + (,, 0) + L -- -(, 0, ) (0, 0, 0) + ^ - ^ (, 0, ) T, s^di.r, + i-i, + i.i) s«x lt x, x )) + saxf, x, x?)) + scat, J*, x%» + ~^-S{{X, X, Z )) + ^ - A s ( ( X *, X*, X*)) (, 0, ) + (0,, ) + (,, 0) + ~ (, 0, ) (0, 0, 0) ^ (, 0, )

9 98] PERFECT MAGIC CUBES OF ORDER k 0 i- (, 0, ) + (0,, ) + (,-, 0) + -^-(, 0, ) (0, 0, 0). + ^(, 0, ) Thus we have found a way of arranging order cubic arrays, in which each of the ordered -tuples on,,, appears exactly once, in an order cubic array B so that in B the coponentwise su of the integers in every row, colun, file, and unbroken diagonal is (0 s 0 s 0). Therefore, as previously stated, we can construct a perfect agic cube of order for odd and >.,D. EKTEHSIOMS AW froblems We know now that there exists a perfect agic cube of order n provided n f,,,, 0,, for odd, and that they do not exist when n,, or. So the question reaining is whether or not there exist perfect agic cubes of orders n,, 0, and s f or odd and >_. It sees probable that such cubes of orders and 0 can be constructed along the lines of Theore. using cubic arrays of orders and that are close to being perfect agic cubes and arranging in the order cubic arrays coposed fro X^ and X% 9 i -,,,, as before. It ay also be possible that by arranging order cubic arrays in order cubic arrays, odd and _>, one can obtain perfect agic cubes of order. As for order, all we know is that if there is a perfect agic cube of order its center is. A ore recent proble in the study of agic cubes is that of extending the into k diensions. For details on this proble and the related proble of constructing variational cubes in k diensions, the reader is referred to [], [], [], [], [], [8], [], and []. REFERENCES. Allan Adler & Shuo-Yen R. Li. "Magic Cubes and Prouhet Sequences." Aer. Math Monthly 8 (9) :'8-.. W. S. Andrews. Magic Squares and Cubes, New York: Dover, 90.. Joseph Arkin. "The First Solution of the Classical Eulerian Magic Cube Proble of Order 0." The Fibonacci Quarterly (9).: -8.. Joseph Arkin, Verner E. Hoggatt, Jr., & E. G. Straus. "Systes of Magic Latin /c-cubes." Can. J. Math. 8 (9): -.. Joseph Arkin & E. G. Straus. "Latin fc-cubes." The Fibonacci Quarterly (9):88-9. W. W. Rouse Ball. Matheatical Recreations and Essays. New York: Macillan, 9.. J. Denes & A. D. Keedwell. Latin Squares and Their Applications. Budapest: Akadeiai Kiado, 9, 8. T. Evans. "Latin Cubes Orthogonal to Their Transpose A Ternary Analogue of Stein Quasigroups." Aequationes Math. 9 (9): Martin Gardner. "Matheatical Gaes." Scientific Aerican ():8- and ():. 0. Ian P.Howard. "Pan-Diagonal, Associative Magic Cubes and -Diension Magic Arrays." J. Recreational Math. 9 (9-):-8.

10 0 GENERATING FUNCTIONS FOR RECURRENCE RELATIONS [April. Ian P.Howard. "A Pan-Diagonal (Nasik), Associative Magic Cube of Order." York University, Toronto, Canada. (Preprint.). Ian P.Howard. "Af-Diensional Queen's Lattices and Pan-Diagonal Magic Arrays." York University, Toronto, Canada. (Preprint.). Maurice Kraitchik. Matheatical Recreations. New York: Dover, 9.. Barkley Rosser & R. J. Walker. "The Algebraic Theory of Diabolic Magic Squares." Duke Math. J. (99):0-8.. Barkley Rosser & R. J. Walker. "Magic Squares, Suppleent." Cornell University, 99. (Unpublished anuscript.). Walter Taylor. "On the Coloration of Cubes." Discrete Math. (9): Bayard E. Wynne. "Perfect Magic Cubes of Order Seven." J. Recreational Math. 8 (9-):8-9. GENERATING FUNCTIONS FOR RECURRENCE RELATIONS LEONARD E. FULLER Kansas State University, Manhattan, KA 0. NTR0VUCTJ0N t In a previous paper [] the author gave explicit solutions for four recurrence relations. The first was a basic relation with special initial conditions. The solution was shown to be related to the decopositions of the integer; n relative to the first positive integers. The second basic relation then restricted the first so that the solution was related to the decoposition of n relative to a subset of the first positive integers. Then the initial conditions for both were extended to any arbitrary values. In the next section we shall give the generating functions for all four of these cases, starting with the initial condition of highest index. We also note the for of the function for arbitrary indices for the initial conditions. Finally, we give a second function that generates all the initial conditions. In Section we give a siple exaple of the fourth kind of relation. We deterine the first few ters of this relation and then copute its generating function. Then we consider relations given in [] and [] and deterine their generating functions.. THE BASIC GENERATING FUNCTION We shall consider a recurrence relation defined by G t Jl r s G t-sl 8- Gi->'--» G o a r b i t r a r y. For notation, we shall refer to its generating function as R (G;x). The first ter generated will be G 0. Later, we shall give a second function that will start with G _. Tke.0A.Jfn.» The generating function for the recurrence relation G n is as follows: R (G; x) L + *,<? _.*» )(l.- E *. * ' ) - To prove that this does generate G n, we set this equal to /~J G n x n and then