# Problem of the Month (May 2004)

A popular type of number puzzle involves placing the digits 1 through 9 into a 3x3 square grid containing some arithmetic operations so that each row and column gives some correct equation. This month we look into the existence of digit isogrids, rectangular puzzle grids using the operations + - x / ^ and the first n positive integers that have the same result in every row and column. Every pair of digits must be separated by an operation, and all operations are performed "left to right" or "top down".

An 3x3 isogrid for 15 exists using only addition: the standard 3x3 magic square. There are 3x3 isogrids for 1, 2, 3, 5, 11, 13, and 18 that use only two different operations. Can you find them? There are 3x3 isogrids for 4, 6-10, 12, 14, 16, 20, and 48 using only 3 operations. The 3x3 isogrid for 36 requires 4 operations. Can you find them?

What results are possible with different sized isogrids? Can you find the possible results in a 2xn isogrid in general? I'm particularly interested in possible results of a 4x4 isogrid. In higher dimensions, what is the smallest isobox?

Guy Segal found a 2x4 isogrid, and used it to prove the existence of 2xn isogrids for every n≥4.

Jordan Balla found a 3x3 isogrid with result 16.

Bill Clagett found all the possible isogrids for the 2x3, 2x4, 3x3, 3x4, and 4x4 cases, including negative and fractional results. Here are his results:

SizePossible Results of Isogrids
2x3-1, [1,3], 7
2x4-4, -2, -1, [1,4], 9
3x3-10, -8, [-6,-1], [1,16], 18, 20, 36, 48
3x4-18, -16, -14, [-12,30], 32, 33, 36, 40, 44, 48, 56, 60, 1/2, 3/2, 5/2
4x4-80, -72, -64, -60, -56, -48, -45, -42, -40, -36, -35, [-33,138], 140, 141, [143,145], 147, 150, [152,154], 156, 157, [159,162], 165, [168,170], 174, 176, 180, 182, 189, 192, 196, 198, 200, 208, 210, 216, 220, 224, 225, 234, 240, 243, 252, 256, 264, 270, 288, 336, 360, 420, 480, 512, 4096, 16777216, -21/2, -15/2, -11/2, -9/2, -7/2, -5/2, -3/2, -3/4, -2/3, -1/2, -1/3, -1/4, 1/6, 1/4, 1/3, 1/2, 2/3, 3/4, 5/4, 4/3, 3/2, 5/3, 7/4, 9/4, 7/3, 5/2, 8/3, 10/3, 7/2, 11/3, 13/3, 9/2, 14/3, 16/3, 11/2, 13/2, 22/3, 15/2, 17/2, 26/3, 28/3, 19/2, 21/2, 32/3, 34/3, 23/2, 25/2, 27/2, 29/2, 31/2, 33/2, 52/3, 35/2, 39/2

This is his amazing 4x4 isogrid with result 224=16777216:

Bill Clagett also noted that there are arrays of numbers which give several different isogrids results depending on the operations inserted. He wondered whether the 3x4 example he found was the smallest one.

Bill Clagett also showed there is a unique 3x3x2 isobox. Below are the two layers. All the signs between the two layers are "-" except for one "^", and all the results are equal to 1.

```    top layer:
1 ^ 18 ^  5
^    -    +
16 -  7 /  9
^    /    -
3 + 11 - 13

bottom layer:
14 - 17 +  4
-    -    +
15 -  6 -  8
^    -    /
2 + 10 / 12
```
Philippe Fondanaiche found all the 3x3 isogrid results. He also showed that for 2xn isogrids, where n≤7, that the results 1-n and 2n+1 were possible. He couldn't extend this to n=8 though, since he couldn't find results of 5 or 6. Can anyone verify this?

Philippe Fondanaiche also found results of 0-34 for 4x4 isogrids using only addition and subtraction.

Here are the small isogrids:

2x3 Isogrids

2x4 Isogrids

3x3 Isogrids
 (Bryce Herdt)

I investigated 2xn isogrids. It turns out results of 2n+1 and some numbers in [1,n] are possible, and nothing in [n,2n] is possible. The following graph shows what exists. the rows indicate n=3 (top) through n=28 (bottom), and the black squares indicate the numbers between 1 and n which are 2xn isogrids.

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 5/25/04.