Problems & Puzzles: Puzzles

Puzzle 1146 Picking out matches. On Sept. 20, 2023, I received a grateful email from an old friend of my pages, Mr. Shyam Sunder Gupta. Here is the main paragraph: I am delighted to share that 18th Sept 2023 paper  “The Guardian” (one of the most popular newspapers in the UK, circulated worldwide), mentions my book “Creative Puzzles to Ignite Your Mind” along with my passion for numbers. Mr. Alex Bellos, one of the great and popular puzzle authors in his biweekly puzzle column in the Guardian (which he has been writing since 2015), called "Alex Bellos's Monday Puzzle" covered some simple puzzles from my book “Creative Puzzles to Ignite Your Mind”. The link is given below which you may like to see. https://www.theguardian.com/science/2023/sep/18/can-you-solve-it-the-man-who-made-indias-trains-run-on-time https://link.springer.com/book/10.1007/978-981-19-6565-4 https://www.amazon.com/Creative-Puzzles-Ignite-Problem-Mathematics/dp/9811965641 BTW, he has his own puzzles web-page here: "Number recreations". Digging over the Shyam's puzzles selected and published  by Mr. Bellos I got captive for number 4, the "Matches Square" one. His puzzle goes like this: Imagine this image is of 40 match sticks arranged to make a 4x4 square.   A total of 30 squares (1 of 4x4, 4 of 3x3, 9 of 2 x2 and 16 of 1x1) can be seen in this arrangement. Find the minimum number of match sticks which on removing, vanishes all 30 squares. (i.e. the perimeter of all of them is broken.) The published solution is this: As usual I started thinkg how can I modify the statement in order to create a "new" puzzle over it. My puzzle is this one: Q. Develop a systematic approach in order to compute and select the picked out matches to satisfy the puzzle condition, for n=8, 9 & 10... Some formulas related to this puzzle: n= rows or columns in the matrix n*n m=quantity of matches needed to construct the matrix nxn = 2*n*(n+1) s = quantity of total squares in the matrix nxn = n*(n+1)*(2*n+1)/6 pm= minimal quantity of picked out matches to interrupt the perimeter of each of the s squares.

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On Dec 8, 2023, Gennady Gusev wen the following contribution:

Consider n x n square.

If n is even, then the number of pairs of squares will be n^2/2. To exclude them, you need to pick out n^2/2 matches. But they are all inside a large square, so we need to remove one match from the perimeter of the large square.

And pm=n^2/2+1.

If n is odd, removing [n^2/2] matches leaves one small square not excluded. So we need one match for small square and one match for large square. Maybe it's the same match. But calculations for small n show that these are different matches. (Unfortunately, I don't have a more accurate proof).

And as total pm=ceiling(n^2/2)+1. (Сeiling function maps x to the smallest integer greater than or equal to x).

n    pm

-    --

4    9

...

8    33

9    42

10    51

11    62

I found a costruction to solve this puzzle and achieve this pm.

For example, I have drawn a 12 x 12 square for clarity. See Fig. 1.

The picked out matches are arranged in 2 triangles (painted wirh grey) of black matches and 2 trianles (painted with pink) of red , as well as steps of the 1 ladder of blue.

As you can see from the picture, it can easily expand to any even number of rows.

If you simply erase the discarded matches, you will get Fig.2.

Interestingly, this construction contains solutions for ALL smaller inner squares.

The solutions for even n begin from the center (see Fig. 3). I circled the borders of the 8 X 8 and 10 X 10 squares in different colors.

The solutions for odd n are shifted from the center by 1 row to the left and up (see Fig. 4). I circled the borders of the 9 X 9 and 11 X 11 squares in different colors.