In each cell of a $7\times 7$ grid is a number. The sum of all the numbers in each $2\times 2$ square and each $3\times 3$ square is zero.

Question

In each cell of a $7\times 7$ grid is a number. The sum of all the numbers in each $2\times 2$ square and each $3\times 3$ square is zero. Prove that the sum of the numbers in the 24 bordering cells is $0$ too.

From which math contest is this problem? MSE has a strict policy about ongoing contests. — , Jan 07 '21 at 11:52
I'm don't actually remember, my teacher mentioned it in passing during class. Maybe the last JBMO? I can't find it in searches or anything so i'm not really sure — , Jan 07 '21 at 11:54
Editing my answer to further clarify: did your teacher also mentioned in passing the last three questions you've asked in this hour? — AnilCh, Jan 07 '21 at 12:02
I am trying out for a test in my home country that picks gifted people and gives them scholarships, you can search it up it's called Qiyas. I finished an online session with my teacher a few days ago where she gave us a list of 27 questions to try and solve - I managed to solve the other 23 but these four I can't find the answer to - sorry if it sounds like I'm ranting, just trying to explain — , Jan 07 '21 at 12:18
Sure, just wanted to be sure. Then I'll comment what I did before: Try with L-shapes. Can you prove that an L-shape of side $3$ sums zero? What another length would you need to finish a proof for the boundary of the $7\times 7$? Do you think you could also prove it also sums zero? — AnilCh, Jan 07 '21 at 12:20

score 2 · Answer 1 · answered Jan 11 '21 at 21:54

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You can split the $24$ bordering cells into 4 groups of $6\times 1$ rectangles and then represent each rectangle as a set difference between an union of two $3\times 3$ squares and an union of three $2\times 2$ squares.

Sometimes a picture worth a thousand words....

answered Jan 11 '21 at 21:54

achille hui

122,701

Very elegant solution! May I ask what software you made the picture with? It looks very clean. – Oussema Jan 12 '21 at 09:47
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@Oussema I have some perl scripts which convert simple geometric objects (including the grid and rectangles) into svg and display it in a browser. I just take a screenshot.... – achille hui Jan 12 '21 at 10:16
Cool. Do you have a github link? Or have you not shared them? – Oussema Jan 12 '21 at 10:19
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@Oussema nope, it is some random personal stuff I developed over the years. – achille hui Jan 12 '21 at 10:20
Got it. Thank you for responding. – Oussema Jan 12 '21 at 10:25

score 1 · Answer 2 · answered Jan 11 '21 at 19:18

Consider a $3 \times 3$ square such as this one.

You can see that we can put a $2\times 2$ square in it (in white here). The sum of the numbers in the "boomerang" shape colored in red is the sum of the numbers in the $3 \times 3$ square minus the sum of the numbers in the $2 \times 2$ square which is zero.

As such the sum of the numbers in every boomerang is zero.

Let's put a boomerang in every corner of our $7 \times 7$ grid below (colored in blue). It just remains to prove that the numbers in the 4 central cells of the grid's sides sum to zero.

Consider the $4 \times 4$ square in the lower left corner. The sum of its numbers is zero because it can be separated into four $2 \times 2$ squares.

However, we can see it can also be separated into a boomerang (in blue), a $3\times 3$ square (in red) and two of the central cells (in green). As such the sum of these two central cells is zero.

The sum of the other two central cells is similarly zero.

As such the sum of the numbers in the border is zero.

In each cell of a $7\times 7$ grid is a number. The sum of all the numbers in each $2\times 2$ square and each $3\times 3$ square is zero.

2 Answers2