Brain Teasers are Back!

I know the feeling Mike ! I blame Ravvie for not reading the question as it was intended. :sunglasses:
I’m glad that JR managed to get his response posted before Ravvie confounded things !

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I suppose I had better have a go at it then. I have an idea how to solve it in a structured way but I think I will need pencil and squared paper for this one.

Away from home this weekend so I will try tomorrow or Tuesday

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Here’s a simpler variation on the multi-size squares teaser. How many are there of each size in a 12x8 grid (ignore the diagonal line).

Bonus point for generalising how many j x j squares are there in a m x n grid?

I found it a useful starting point.

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Counting Squares gives 96, 77, 60, 45, 32, 21, 12 and 5

Now, if only there was a ‘pattern’ to these numbers …

(of course there is one) :sunglasses:

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Something like (m+1-j) x (n+1-j)

There could be quite a few patterns I guess.

One I noticed in passing is that ‘Starting’ with 0, add 5 = 5

Then add the next odd number, ie 7, to give 12
Then add the next odd number, 9, to give 21
Then 11 to give 32 etc etc.

But the real ‘pattern’ that I was working with was much more relevant and involves the successive reduction by ONE from 12x8; 11x7 …

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Yes, that’s the pattern I was thinking of. Bonus point goes to Don.

Can you explain why the pattern is so?

Let’s try this …

Draw a 2x2 square in the top left-hand corner of the 12x8 rectangle.
This can be moved successively, one small square at a time, 11 times to the right.
It can also be moved successively, one small square at a time, 7 times down.
Hence a 2x2 square can occupy 11x7=77 different locations within the 12x8 rectangle.

Increase the ‘moveable’ square by 1x1, to make a 3x3 'moveable square.
This can only be moved 10 times to the right and 6 times down.

ie increase the ‘moveable’ square by 1x1, and the number of positions it can occupy decreases by 1x1.

Please note however, that a set of 3x3 squares can’t completely fill a 12x8 rectangle (without overlapping).

1x1 squares can. Some 2x2 squares can. Some 4x4 squares can. The rest, 3x3, 5x5 etc, can’t.

96 (1x1)
24 (2x2)
6 (4x4)



That’s a good clear explanation of how 12x8 reduces to 11x7 etc. I have used these, plus your general formula to solve the extended teaser.

Mrs R thinks that I have missed some squares, but she is still to look at it properly. As usual, we have used two very different approaches.

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Let’s rely on Mrs R for the definitive answer.

My eyes seem to have gone square-shaped, but I appear to be going round in circles !!

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These represent the number of squares that (I think) can fit within a 12x8 rectangle and completely fill it without overlaps.

The number of such squares that a diagonal of the rectangle would pass through will, of course, be less. My current estimate is 28, including JR’s initial 16.

Ah! Another interpretational difference. I have included overlaps, on the basis that they are still squares within the overall grid.

Ok, I’ll give it a go with the overlap interpretation.

Working from 1x1 thru 8x8 I get:- 16, 29, 36, 37, 32, 21, 12 and 5 making a total of 188

There is a degree of symmetry in the construction of each of these numbers, again working from 1x1 thru 8x8 …
2, 2, 2, 2, 2, 2, 2, 2
3, 5, 4, 5, 4, 5, 3
5, 6, 7, 7, 6, 5
6, 8, 9, 8, 6
8, 8, 8, 8
7, 7, 7
6, 6

Other symmetries are available !! :sunglasses:

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Nicely set out, Don, and the results agree with mine. Mrs R has been busy preparing her Open University students for a September exam, so that has taken priority. She only had a brief look earlier.

I used two approaches - I think my second approach is equivalent to yours, though I wrote a short program in Python to avoid my eyes going round in circles (or is that squares?) It gave answers for any m x n rectangle, though disappointingly I couldn’t spot any obvious patterns in the totals.

My first approach is a bit different, using a bit of maths to minimise having to go through every possibility manually. Though given the symmetry, I imagine that you only had to consider half of them (or half of your top 4 rows).

I will post my answer when I get back home tomorrow night.

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Nothing subtle at all. Just a plain, old-fashion slog. Well, it only took a few minutes and the symmetry helped with the check. But I did check three of them anyway !

Nice work! I see you also made use of the formula you found to do the checks.

Sorry for inadvertently creating a harder teaser!

Mrs R had a go at the 2x2 squares and agreed 29. She also methodically slogged through. She started with non-overlapping squares and then shifted a virtual overlay into a total of 4 positions to give 8+6+8+7=29. I asked how she would do 3x3 and 4x4. She noted that this would need 9 and 16 positions, so would be a bit laborious. Maybe not so bad if she still had those clear acetates that she used to use for overhead projector totorials!

My method still to follow…

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I found it easier to count those that the diagonal didn’t cross then subtract from the total (using the equation). Consider the red shape in the diagram and double it for the equivalent shape above the diagonal:

I split into 3 sections. Need to determine how many squares wholly within A, B and C and also those that intersect, i.e. cross a dotted line.

A: Easy, just use the formula
B: Count manually
C: Same shape as B so no further work
Intersections: Count manually


They are consistent with Don’s. I only needed to manually count 7+10+3=20, so not too much eyes glazing over the grid!

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… well. the nights are drawing in, and the stars are shining brightly, so …

2/5 + 1/7 = 19/35?


Same result here, as 2/5=14/35 and 1/7=5/35.