Mean Square
Problem
Can you fill this 3 by 3 grid with the numbers 1 to 9, so that the number in the middle of each line (horizontal, vertical and diagonal) is the mean of the two numbers on either side of it?
Student Solutions
The mean of 2 numbers will always be between the 2 numbers.
Whichever number goes in the middle will need to be the mean of the most different pairs of numbers, so it must be the 5, because the 5 is in the middle of all of the numbers.
| 5 | ||
Then the numbers which go either side of the 5 must be pairs of numbers that add up to 10 (because to find the mean of 2 numbers, they must be added together and then divided by 2). So they must be:
1 and 9
2 and 8
3 and 7
4 and 6
9 cannot be the mean of two of the other numbers, because it is the largest number. So it must go in one of the corners.
| 1 | ||
| 5 | ||
| 9 |
9 is odd, and if it is added to an even number, then the result will be odd. When this odd number is divided by 2, the result will not be a whole number. So the numbers in the other corners cannot be even numbers. So they must be 3 and 7.
| 1 | 3 | |
| 5 | ||
| 7 | 9 |
Filling in the remaining boxes gives the completed square.
| 1 | 2 | 3 |
| 4 | 5 | 6 |
| 7 | 8 | 9 |
Other solutions can be obtained by rotating and reflecting this one.