To warm up, let's start with $k=3$. We want to prove that the numbers within the large square that's $nth$ from the left ad $mth$ from the top sum to $$(2m-1)(2n-1)T_3^2=(2m-1)(2n-1)9.$$ First look at the large square that's $nth$ from the left and lies in the first row of large squares. As we already noted above, the two numbers in the top row of this white square are $3(n-1)+1=3n-2$ and $3(n-1)+2=3n-1.$ By some very similar reasoning we can work out that the top two numbers in the large square that's $nth$ from the left and $mth$ from the top are $$(3(m-1)+1)(3n-2)=(3m-2)(3n-2)$$ and $$(3(m-1)+1)(3n-1)=(3m-2)(3n-1)$$ Similarly, the bottom two numbers in the $nth$ white square from the left in the $mth$ row of white squares are $$(3(m-1)+2)(3n-2)=(3m-1)(3n-2)$$ and $$(3(m-1)+2)(3n-1)=(3m-1)(3n-1).$$ Adding these four numbers gives $$\begin{array}{cc}& (3m-2)(3n-2)\\ +& (3m-2)(3n-1)\\ +& (3m-1)(3n-2)\\ +& (3m-1)(3n-1)\end{array}$$ which we can rewrite as the product $$(3m-2+3m-1)(3n-2+3n-1)=(6m-3)(6n-3)=(2m-1)(2n-1)9,$$ which is exactly what we wanted. We now move on to prove the result for any positive integer $k$. We want to show that the sum of numbers in the square that's $nth$ from the left and $mth$ from the top is equal to $$(2n-1)(2m-1)T_{k-1}^2.$$ Let's call the square in question $T.$ Using similar reasoning as above, you can see that the numbers in the first row of $T$ are $$(k(m-1)+1)$$ times the numbers in the first row of $S$. This means that the sum of the first row of numbers in $T$ is $$(k(m-1)+1)(2(n-1)T_{k-1}).$$ Similarly, the sum of the second row of numbers in $T$ is $$(k(m-1)+2)(2(n-1)T_{k-1}).$$ We can continue in this vein until we come to the sum of the last row of numbers in $T,$ which is $$(k(m-1)+(k-1))(2(n-1)T_{k-1}).$$ Adding up these sums of the rows of $T$ gives $$\text{Missing begin{equation} or extra end{equation}}$$ Similarly to what we did above, we can re-write this expression as $$[1+2+...+(k-1)+(k-1)k(m-1)][(2n-1)T_{k-1}].$$ Using the formula for the sum of the first $k-$ integers, this becomes $$[\frac{(k-1)k}{2}+(k-1)k(m-1)](2n-1)T_{k-1},$$ which is equal to $$\frac{(k-1)k}{2}(2m-1)(2n-1)T_{k-1}.$$ And since $$\frac{(k-1)k}{2}=T_{k-1},$$ the sum of the numbers in $T$ is equal to $$(2m-1)(2n-1)T_{k-1}^2.$$ This is what we wanted to show.

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