7. arithmetic & number theoretic recreations a. Fibonacci numbers

Prob. 3, p. 2. Gives triangle with sides 2, 5, 4; 4, 1, 6; 6, 3, 2.

Prob. 4, p. 2. Gives triangle with sides 8, 1, 6, 5; 5, 4, 9, 2; 2, 3, 7, 8. Gives a number of simple consequences of the magicness and also that the sum of the squares of the numbers on a side is 126. [There are 18 magic triangles, but only this one has the sum of the squares constant.]

Collins. Book of Puzzles. 1927.

Pp. 92-93: The magic triangle. Consider a triangle with two points between the vertices. Put the numbers 1 - 9 on the vertices and intermediate points so that the sum of the values on each edge is constant and the sum of the squares of the values is constant. Gives one answer and is somewhat vague as to whether it is unique. See Peano.

P. 93: A nest of magic triangles. Says this occurs on a 1717 document of the Mathematical Society of Spitalfields. Start with a triangle and join up its midpoints. Repeat on the resulting triangle and continue to the fourth tie, getting five triangles with 18 points. The values 1 - 18 are placed on these points to get various sums which are multiples of 19.

Collins. Fun with Figures. 1928.

The Siamese twin triangles, pp. 108-109. Triangle with 4 cells along each side. Place the digits 1 through 9 on the cells so each line adds to 20. Two complementary solutions, with sums 19 & 21. He gives a number of further properties about various sums of squares.

A magic hexagon within a circle, pp. 110-112. This is really a pattern of six magic triangles like the above, with sums 20 and with sums all distinct, with a further property about sums of squares.

Perelman. FFF. 1934. 1957: probs. 46-48, pp. 56-57 & 61-62; 1979: probs. 49-51, pp. 70 71 & 77-78. = MCBF, probs. 49-51, pp. 69-70 & 74-75.

49: A number triangle. Triangle with 4 cells along each side. Place the digits 1 through 9 on the cells so each line adds to 20 -- as in Collins, pp. 108-109. One solution, but he notes that the two central cells in each line can be interchanged.

50: Another number triangle. Same with total 17, again one solution given.

51: A magic star. Star of David with cells at the star points and the intersections. Place numbers 1, .., 12 so each line of four and the six points all add to 26. One solution given.

Ripley's Puzzles and Games. 1966. Pp. 34-35, item 2. P;ace the fifteen pool balls so each edge and the central three balls total to the same sum. Ripley's gives examples with magic sum of 34, 35, 36. One can construct examples with magic sum of 32, 33, ..., 39. A sum of 40 initially seems possible but further analysis shows it is not possible. If one rules out trivial rearrangements leaving the rows having the same elements, there are 2716 distinct solutions. Each of these has 1296 trivial rearrangements.

Jaime Poniachik, proposer; Henry Ibstedt, solver. Prob. 1776 -- Connected differences. JRM 22:1 (1990) 67 & 23:1 (1991) 74-75. Triangular lattice with edge 2. Place numbers 1, ..., 15 on the 6 points and 9 edges so that each edge is the difference of its end points. 19 solutions found by computer. [Not sure where to put this item??]

7.N.3. ANTI MAGIC SQUARES AND TRIANGLES
An antimagic n x n has its 2n+2 sums all distinct. A consecutively antimagic n x n has its 2n+2 sums forming a set of 2n+2 consecutive integers. Berloquin calls these heterogeneous and antimagic, respectively. A heterosquare has all the 4n sums along rows, columns and all broken diagonals being different.
Loyd Jr. SLAHP. 1928. Magic square reversed, pp. 44 & 100. 3  2  7;   8  5 9; 4 6 1 has all eight sums different.

Dewey Duncan. ?? MM (Jan 1951) ??NYS -- cited by Gardner. Defines a heterosquare as an arrangement of 1, 2, ..., n² such that the 4n sums along the rows, columns and all broken diagonals are all different. (However, in the next item it appears that only the main diagonals are being considered??) Asks for a proof that the 2 x 2 case is impossible and for a 3 x 3 example -- which turns out to be impossible.

John Lee Fults. Magic Squares. Open Court, La Salle, Illinois, 1974. On p. 78, he asserts that Charles W. Trigg posed the problem of non existence of anti magic 2 x 2's in 1951, that it was solved by Royal Heath and that Trigg gave the spiral construction for anti magics of odd order. Unfortunately Fults gives no source, only noting that Trigg was editor at the time. There is nothing in Heath's MatheMagic. I now see that this is a corruption of the preceding item. Madachy (see below at Lindon) refers to the problem in MM (1951) without further details, but restricted to just the main diagonals, and then says "An exchange of correspondence between Charles W. Trigg, then "Problems and Questions" editor for the magazine, and the late Royal V. Heath, ..., soon established some basic properties of potential heterosquares."

C. W. Trigg, proposer; D. C. B. Marsh, solver. Prob. E1116 -- Concerning pandiagonal heterosquares. AMM 61 (1954) 343 & 62 (Jan 1955) 42. The solution is also in: Trigg; op. cit. in 5.Q; Quickie 160: Pandiagonal heterosquare, pp. 45 & 151. There is no arrangement of 1, 2, ..., n² such that the 4n sums along the rows, columns and all broken diagonals are consecutive numbers.

Charles F. Pinska. ?? MM (Sep/Oct 1965) 250-252. ??NYS -- cited by Gardner. Shows there are no 3 x 3 heterosquares, but gives two 4 x 4 examples.

Gardner. SA (Jan 1961) c= Magic Numbers, chap. 2. Notes that 1  2  3;   8 9 4; 7 6 5 is anti magic, i.e. all 8 sums are different, and it is also a rook's tour. In Magic Numbers, Gardner says he had not known of anti-magic squares before seeing this one, but later discovered the Loyd example. He summarises the knowledge up to 1971.

J. A. Lindon. Anti magic squares. RMM 7 (Feb 1962) 16 19. Summarised and extended in Madachy; Mathematics on Vacation, op. cit. in 5.O, (1966), 1979, pp. 101-110. Author and editor believe this is the first such article. Wants the 2n+2 sums for an n x n square to be all different and also to be a set of consecutive integers. There are no such for n = 1, 2, 3, but they do exist for n > 3.

M. Gardner. Letter. RMM 8 (Apr 1962) 45. Points out his SA article and notes that 9  8  7,   2 1 6, 3 4 5 is even more anti magic in that the 8 lines and the 4 2 x 2 subsquares all have different sums.

Pierre Berloquin. The Garden of the Sphinx, op. cit. in 5.N. 1981.