Mean Square
Can you fill this square so that the number in the middle of each line is the mean of the two numbers on either side of it?
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.