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Math and Logic

Sweet Sixteen

Given the integers 11 through 99 in a 3×33 \times 3 grid, we can rearrange those integers so that every row and column has a sum of 15:15:

This is just one example; there are more arrangements that work. This is just one example; there are more arrangements that work.

Suppose that, instead, starting with that same initial arrangement of 11 through 99 in a 3×33 \times 3 grid, we replace any number of the integers with 00 to try to make every row and column have the same sum.

This problem has no solution ((other than the trivial solution of replacing every integer with 0).0). To see why — and why for a problem like today’s challenge — keep reading. Otherwise, jump ahead to the challenge.

As arranged, the sum in the top row of the grid is 6.6. To make all the rows and columns have the same sum by replacing integers with 00s, each row and column would need to have a sum of 66 or less, to match that top row. Looking at the bottom row, the only way to make it have a sum of 66 or less is by making it have a sum of 0.0. Then every integer in the grid would have to be replaced with 0.0.

In today’s challenge, the starting 5×55 \times 5 grid is filled with integers, but they aren’t 11 through 2525 in order and some of the integers repeat. Here we’ll work through a similar example with a 4×44 \times 4 grid:

Given this grid, the goal is to replace any of the integers with 00 so that the sum of every row and column is 12.12.

The most challenging part can be deciding where to begin. Try to find the most restricted part of the grid where a row or column can only sum to 1212 in one or two ways. Consider the second row: the only way to make a sum of 1212 with 5,2,10,5, 2, 10, and 44 is 2+10.2+10. So the 55 and the 44 need to be replaced with 00s:

Now we know that the 1010 and 22 in the second row cannot be replaced with 00s. To make the third column with that 1010 in it sum to 12,12, we must replace the 77 and 33 with 00s:

In the second column, we need each one of 7,2,2,7, 2, 2, and 11 to sum to 12.12. So none of those get replaced with 00s. In particular, the bottom row must use that 11 in its sum. Then we must also use the 22 and 99 in the bottom row, while the 1212 must be replaced with a 0.0. Meanwhile, in the top row, one of the 55s must be replaced with a 0.0. The 55 in the left column we need, the 55 in the right column we don’t need.

By replacing six of the numbers with 00s, we now have every row and column sum to 12.12. Can you do the same with a slightly larger square and target sum?

Today's Challenge

In this challenge, you can replace any number in the grid below with a 0:0:

How many numbers in the grid must be replaced with 00s to make the sum of every row and column 16?16?

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