which on Dürer’s square do not add to 34.

My square’s first line is 8 11 14 1

second line 13 2 7 12

third 3 16 9 6

fourth 10 5 4 15

Add for example the four top left corner: 8+11

+ 13 + 2 = 34

Dürer’s top corner is: 16 + 3

+ 9 + 10 = 38

I will appreciate comments on this work. ]]>

Great explanation Uri:)

כל לחי שרון

]]>A square of size n contains all the integers in the range 1..n².

The sum of the integers in the range 1..m is given by m(m+1)/2, and dividing this by m to get the average, gives us: (m+1)/2.

Now substitute n² for m, and we get that the average of the numbers in an n-sized square is: (n²+1)/2.

For an (n-1)-sized square, we replace n by n-1, giving us:

((n-1)²+1)/2 = ((n²-2n+1)+1)/2 = (n²-2n+2)/2.

Subtracting this from the value we found for the n-square gives us:

(n²+1)/2 – (n²-2n+2)/2 = (n²+1-(n²-2n+2))/2 = (2n-1)/2.

Which happens to be exactly the average of n and n-1:

(n+(n-1))/2 = (2n-1)/2.