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Different Totals

Problem

Using each of the digits 1 to 5 once, it is possible to place them in the grid so that the row and column have the same total.


How many different totals can this be done with?


Solution

Begin by listing all possible totals using three digits from 1, 2, 3, 4, 5:

   Total
1236
1247
1258
1348
1359
14510
2349
23510
24511
34512

So it is possible to fill the grid with three different totals:

8 = 2 + 5 + 1 = 1 + 3 + 4
9 = 5 + 1 + 3 = 3 + 2 + 4
10 = 1 + 4 + 5 = 5 + 2 + 3
   i.e.

Alternatively, note that the common square in the top right corner is always odd... The sum of all five digits, 1 + 2 + 3 + 4 + 5 = 15. If an even digit is placed in the common square: 15 minus 2 = 13 or 15 minus 4 = 11, the remaining total will be odd, and this cannot be split between the two legs equally. Hence the only way in which this can be completed is to place an odd digit in the common square: 15 minus 1 = 14 (7 on each leg), 15 minus 3 = 12 (6 on each leg) and 15 minus 5 = 10 (5 on each leg).

What about using the digits 1 to 7 with the following grid?


Problem ID: 85 (Nov 2002)     Difficulty: 1 Star

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