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In a blog post I see a multiplication chart:

enter image description here

The post then says:

...can you see why summing the first $n$ odd integers results in $n^2$?

But I can't see it. Can someone help me understand why summing the first $n$ odd integers results in $n^2$?

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  • $\begingroup$ Hint: What is the sum of the first $n$ integers? $\endgroup$ – Matti P. May 12 '17 at 9:46
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    $\begingroup$ While it is true that this was already proved elsewhere, here it seems that the OP is looking for an explanation about how one can get the result looking at the picture $\endgroup$ – lesath82 May 20 '17 at 18:44
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For each colour, there is an odd number of squares coloured in that colour. Their cumulative sums give you squares.

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  • $\begingroup$ "Each colour represents an odd number" - Please can you explain this? I presume you mean an odd number of squares, but cannot connect this to the solution. $\endgroup$ – Ben May 12 '17 at 9:32
  • $\begingroup$ @Ben Because the number of blocks in those stripes is $\text{height}+\text{length}-1=2\times\text{height}-1$ and $\text{height}=1,2,3,\cdots$. $\endgroup$ – user228113 May 12 '17 at 9:34
  • $\begingroup$ for each number, 1,2,3.. there is a square you can see in the top left, with $n^2 $ blocks in it - when you go to the next number, you have to add another shell, which has 2 extra blocks in it - for the number 4 for example, the outer shell has 2 sets of 4, plus 1 in the corner - so we added 7 to get 16, the previous number was 9, the square of 3 $\endgroup$ – Cato May 12 '17 at 9:45
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sum of squares

Above is a picture that proves $1+3+5+7+9 = 5^2$

If you look at the table, you see that the numbers in the diagonal are $1^2, 2^2, 3^2, 4^2, \dots$. Now look at the way the cells are colored.

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  • $\begingroup$ this pictures is convincing but how do you connect it with the multiplication chart? $\endgroup$ – Nathanael Skrepek May 12 '17 at 9:55
  • $\begingroup$ the chart has same coloured areas, which correspond to the areas marked by heavier lines in the picture here. $\endgroup$ – Cato May 12 '17 at 10:11
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Looking at your colored chart, start at $1$, and you obviously have $1^2$. Moving to the next color you add $3$, that is the previous $1$ plus $2$, and you build the next square: $2^2$. The next adds the previous $3$ plus other $2$ and you get $3^2$. The rule is that at each step you are adding the next odd number (the colored cells are always $2$ more than in the previous step) and you are making the square's side longerr by $1$.

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