1.17 Triangle Numbers

Transcription

1 .7 riangle Numers he n triangle numer is nn ( ). he first few are, 3,, 0, 5,, he difference etween each numer and e next goes up y each time. he formula nn ( ) gives e ( n ) triangle numer for n. he n triangle numer is e same as n C riangle numers can e illustrated wi equilateral or right-angled isosceles of dots. We include ecause it completes e pattern of differences and fits e aove formulas. For e same reason, we say at is e first square numer, since =...7. Handshakes. If everyone in e room were to shake hands wi everyone else in e room, how many handshakes would ere e? Start small. ry 3 people. Stand up and do it. Now try a group of 4. Someody keep count. An alternative context is sending Christmas cards (or similar) if everyone sends one to everyone else how many cards will ere e? his avoids e prolem of doule-counting, since if A sends a card to B, B (out of politeness!) will send a card to A. You can do is wi teams playing matches if you say at every team wants to play every oer team at home..7. Rectangles. How many rectangles (of any size) are ere in is shape? here are more an 4. Can extend to dimensions (i.e., like a chessoard). Here, a square counts as a rectangle..7.3 Straight lines and intersections. How many crossingpoints are ere when 4 lines overlap if each new line If ere are n people, each of em needs to shake hands wi everyone except emselves; i.e., n people, so at makes nn ( ), ut is counts every handshake twice from o ends. So e answer is nn ( ). Each person has to send n cards, ecause ey send one to everyody except emselves. here are n people at do at, so nn ( ) cards altogeer. If n teams all play each oer at home, en all n venues are visited y all n oer teams (oviously ey don t play against emselves!) and at is just nn ( ) games. Answer: 0, e 4 triangle numer You can explain it y inking aout what happens when you add anoer lock on to e right end to make 5 locks. he 5 lock makes 5 more rectangles: itself, itself and e one to its left, itself and e two to its left, itself and e ree to it s left, and itself and e four to its left. his happened ecause ere were already 4 rectangles ere. In general, adding e n rectangle increases e total y n. An array of rectangles x y will contain a total of xx xy x 4 ( ) yy ( ) rectangles ( )( y ). his is ecause e first row contains xx ( ) different rectangles, and each of ese has height, so y( y ) of em can e fitted vertically down e grid. If x y (a square array, like a chessoard), is reduces to x ( x ), so when x 8 (for a 4 chessoard), e total is 9 rectangles (e sum of e st eight cue numers). (See section.4..) Need to use A4 paper at least and choose e angles of e lines wisely. Colin Foster, 003

2 is drawn so at it crosses as many lines as possile? intersections How many regions are produced at each stage? (Count line as producing regions.) A context for is is cutting up a cake: What is e maximum numer of pieces you can divide a cake into using 5 straight cuts? he pieces don t have to e equal sizes. Start small try,, 3 cuts first..7.4 Baked Bean ins. hese can e stacked in a triangular array, giving e triangle numers. his is easy to extend to tetrahedron numers y making a triangular horizontal layer and putting a smaller triangular layer on top until you reach one can at e very top. he finished stack is roughly tetrahedral. It s easy to miscount or draw lines which don t cross e maximum possile numer of lines. he n line should cross n lines, so after e n line ere should e nn ( ) intersections. (Every line crosses n oers, making nn ( ) crossings, ut is doule-counts every line, so we put in a factor of.) Or you can say at ere will e a crossing-point for each pair of lines, so it s e numer of ways of n nn ( ) choosing from n, or C, which is e! same. regions = pieces of cake after n cuts = ( n n ) n( n ) ; i.e., one more an e n triangle numer. See a similar investigation (aout e maximum numer of formed from intersecting lines) in section.9.. he n tetrahedron numer is e sum of e first n triangle numers, so ey go, 4, 0, 0, 35, 5, 84, ;; in general e n one is nn ( )( n ). Why does is formula always give a positive integer whenever n is a positive integer? Answer: Because n, n and n are consecutive integers. his means at eier one of em or two of em must e even, so when you multiply all ree you re ound to get a numer at is divisile y. Also, one of em will always e a multiple of 3 (ecause multiples of 3 come every ree numers), so e product will e divisile y 3. So when you divide y you get an integer. Making each layer a square array of cans instead, gives e square pyramidal numers. he n square pyramidal numer is e sum of e first n square numers, so ey go, 5, 4, 30, 55, 9, 40, ;; in general e n one is nn ( )(n ). Why does is formula give an integer value? Answer: Eier n or n must e even, so e product of e ree numers must e divisile y. If eier of ese numers is a multiple of 3, en of course it will e fine. If neier is, en n must e of e form 3m, where m is an integer, and at means n 3m and n (3m ) m 3, which is divisile y 3. So one of e numers will always e divisile y 3, and one divisile y, so dividing y must give an integer answer. Colin Foster, 003

3 .7.5 How many small are ere altogeer in is drawing? Anoer way to count em is row-y-row. his shows at e sum of e first n odd numers is e n square numer. Because e large equilateral triangle is maematically similar to e small ones, as e numer of rows increases as n e area (e numer of small equilateral ) increases as n. How many (of any size) are ere altogeer in e drawing? (his is much harder.) he sum of e st n triangle numers is e n tetrahedral numer nn ( )( n ) (see section.7.4). For n even, e numer of type is nn ( )(n ). 4 For n odd, e numer of type is 4 ( n )( n n 3). Adding nn ( )( n ) to each of ese gives e formulas for e total numer of when n is even and when n is odd. Answer: 5. Start wi fewer rows and look for a pattern. Can count wi a side at e ottom (type ) and wi a point at e ottom (type ) separately (shade in one set). rows of (type ) (type ) total no. of so if ere are n rows of small, e total numer of = n. So we see at e sum of two consecutive triangle numers is a square numer, or nn ( ) ( n ) n nn ( n ) n n or n + n = S n. Answer: 48. Again, start wi a drawing wi fewer rows and look for a pattern. rows of (type ) (type ) total no. of After n rows e numer of type is e sum of e first n triangle numers. he numer of type is more complicated, ecause you only get a igger type triangle on every oer row (a igger type triangle appears wi every new row). So e numer of type depends on wheer n is even (en it s e sum of e alternate triangle numers starting wi e first one ()) or odd (en it s e sum of e alternate triangle numers ut eginning wi e second one (3)). So e total numer of is nn ( )(n ) 8 if n is even, and 8 ( n )( n 3 n ) if n is odd..7. Which integers from to 0 can e made from e sum of just triangle numers? (he triangle numers emselves can oviously e Answer: All except 5, 8, 4, 7 and 9. Colin Foster, 003

4 made from just one triangle numer.) Gauss ( ) proved at every integer is e sum of at most ree triangle numers. Which numers can e expressed as e sum of two consecutive triangle numers?.7.7 Polygon Numers. Can you define pentagon, hexagon, heptagon, etc. numers? Find different formulas for e different n polygon numers. his can also e written as nnp ( n p 4). What kind of numers are rectangle numers? If you put p into e formula you just get n..7.8 Mystic Rose. Space four points evenly around e circumference of a circle. Join every point to every oer point. How many lines do you need? Wi more points it makes eautiful patterns at are suitale for display work..7.9 When we use e formula nn ( ) (where n is an integer), why do we always get an integer answer? (Picking just any two integers and multiplying em and dividing y won t necessarily give an integer.).7.0 Prove at 8 times any triangle numer is less an a square numer. no. sum no. sum no. sum = = + 3 = 3 4 = = = 7 = + 8 = = = 0 = 0 + = + 3 = = = 5 + = = = = = and oer possiilities. 4, 9, ; i.e., e square numers (see section.7.5). Answer: Make e leng of each side increase y as n goes up y. If p is e numer of sides of e polygon (so p 3 for e triangle numers), en e n p - gon numer is nn ( ( p ) ( p 4)), and for n is reduces to no matter what e p value. So is e first numer in all e polygon numer sequences. Rectangle numers would e eier all e integers (if you allow rectangles wi a side of leng ), or all non-prime numers (if you don t). (Non-primes are called composites.) Answer: (four sides of e square plus two diagonals) For n points, e numer of lines is e ( n ) triangle numer, nn ( ). his happens ecause each of e n points is connected to e oer n points, making nn ( ) lines, ut is doule-counts each line (from o ends) so we put in e factor of. Finding e numer of regions is deceptively difficult see section.9.. n and n are consecutive numers, and at means at one is always odd and e oer even. herefore you re always multiplying an odd numer y an even numer, and at always gives an even answer (multiplying any integer y an even numer always gives an even answer). So when you halve e answer you get an integer. Anoer way to ink aout it is at n ( n ) nn ( ) ( n ) n ; i.e., you can halve eier n or n (whichever is even) and multiply e answer y e oer one, so you ll always get an integer. Answer: 8 nn ( ) 4 nn ( ) 4n 4n, which is less an odd square. (n ), so it s always less an an Colin Foster, 003

5 .7. Consecutive Sums. What is e total of all e integers from to 00? What aout to 000? Or more? You can tell e story of Gauss ( ) who was told y a teacher to add up all e numers from to 00 (wiout a calculator, of course, in ose days) and did it very quickly. Pupils may ink at it ought to e possile to replace repeated addition y some sort of multiplication, and at is what we re doing. x x Which integers is it possile to make using e sum of consecutive positive integers? Answers: S0 55, S and S , etc. (where S means e sum of all e integers from a a to inclusive; i.e., i ). i a Various approaches:. Realise y drawing dots or squares, at S x is e x triangle numer. Use e formula xx ( ) or see is y comining two identical of dots and getting a rectangle x y ( x ).. Say at e average value (actually e mean and median) of e numers from to x must e x e, and since ere are x values e total must x( x ). 3. Pair up and x (to make x ), and x (to make x also), etc. Eventually you have x pairings (if x is even), so e total is x( x ) x (But what if x is odd? hen ere are pairings and e middle numer ( x over. So e total is time is x x x x ( ) ( x ) x( x ). ) is left, e same.) 4. Find S0 55 y some meod (or just add em up) and argue at S 0 must e 00 more, ecause each numer in e sum is 0 more an each numer in e first sum (writing out some of it makes is clearer). So altogeer S must e 0 0 igger 55. Now 0 S , so S (y similar reasoning), and so S 40 80, and so on, giving S , S80 340, S ( ), so S S Alough is meod is not quick, it does involve some good inking. See sheet Sums of Consecutive Integers. Clearly all odd numers (except ) are possile (see diagonal line in e tale), since pairs of consecutive integers added togeer make all of e odd numers. Impossile totals are,, 4, 8,, (powers of ). You can see is ecause in e formula ( a)( a ), one racket must e a even and one odd, so a has at least one odd factor, and so can t e a power of. Colin Foster, 003

6 Sums of Consecutive Integers he sums of consecutive integers are e trapezium numers, which are e differences etween (non-consecutive) triangle numers. a his happens ecause x is e a triangle numer ( a ) and x is e triangle numer ( ). [ n n( n ) ] So x a x x x (e sum of e consecutive integers from a to ) is a, where a 0 and a. [ a ( a)( a ) ] he possile integer sums are given in italics in e tale elow (horizontal triangle numer minus vertical triangle numer)