4.8.3 Understanding the Puzzle
Add up the numbers in the top row. This is .
Now, instead of picking the five numbers from the top row, we pick one from each of the rows in turn, selecting one from each of the columns.
- The number we pick from the first row is one of the five in the sum above. Suppose we pick 3.
- The number we pick from the second row is in the same column as one of the numbers above and is 5 more than it. For instance, if we choose the column headed by 2, then the number we get is .
- The number we pick from the third row is under a different one again of the five numbers above, and is 10 more than the one in the top row. Say we choose the column headed by 5 then the number we get is .
- The same goes for the fourth and fifth rows, with 15 and 20 more than in the first row, respectively.
So, compared with picking all five numbers from the top row, choosing one number from each row, with the constraint that each must also be in a different column, means that their sum will exceed the sum of the numbers in the first row by 5 (for the second row rather than the first) + 10 (for the third row rather than the first) + 15 (for the fourth row rather than the first) + 20 (for the fifth row rather than the first).
Now, , so it will be 50 more than 15 (the sum of the numbers in 1st row). Now to find the total we need to add back in the sum of the first row numbers and we have .
So whatever numbers you pick the sum will always be 65.