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#22 Even rows

Set by Peter Rowlett

Fill in the numbers 1-16 once each in a four-by-four grid so that the sum of the numbers on the top row is even and the numbers on the bottom row sum to a different even number.

For example, if we arrange the square so that the numbers across the top row are 1, 2, 3 and 4, they will sum to an even number, 10.

Can you find a solution where all four edges have different sums that are all even?

Can you find a solution where all rows, columns and diagonals sum to different even numbers?

Solutions next week

#21 Digit gangs

Solution

For the largest possible sum of ABC + DE + FGH + IJ, we need to assign the two largest digits (9 and 8) to the hundreds digits, A and F, then the next largest to the tens digits, making sure the four smallest digits are in the units position. For example, 973 + 51 + 862 + 40. It doesn’t matter what order you arrange these in; the sum will always be 1926. To get the largest outcome from (ABC × DE) + (FGH × IJ), we can maximise one of the two products by using the 8 and 9 as the first digits, so the largest result is (851 × 94) + (630 × 72) = 125354. Also this can be achieved in a few other ways: (720 × 63) + (851 × 94), (851 × 94) + (630 × 72) and (851 × 94) + (720 × 63). There are 198 ways to get a difference of exactly zero from (ABC × DE) – (FGH × IJ), including (135 × 96) – (270 × 48) and (972 × 15) – (486 × 30).