Daily Maths Challenges

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siriusmart

Find a+b (self.dailymaths)

submitted 6 months ago* (last edited 6 months ago) by zkfcfbzr to c/dailymaths

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The image is of a large unit square with five smaller disjoint shaded squares contained entirely within it. The five smaller squares are congruent. Four of them are at each corner of the large square. The fifth is in the center, rotated diagonally, so the center of each of its sides is touched by the vertex from one of the other four squares. You are given that the common length for the five smaller congruent squares is (a-sqrt(2)) / b, where a and b are positive integers. What is the value of a + b?

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[–] siriusmart 4 points 6 months ago (1 children)

spoiler

https://gmtex.siri.sh/fs/1/School/Extra/Maths/Challenges%20solutions/find-a+b.html

[–] zkfcfbzr 3 points 6 months ago* (last edited 6 months ago) (1 children)

All good 👍

spoilerish commentary

This is the short / simple solution I mentioned in my other comment. I think when I originally encountered this problem (well over a year ago) I tackled it by making an equation that added to 1, the side length of the unit square - but picking the diagonal makes everything work out far, far simpler.

Also, not sure if you care, but that link allows full access to everything in your drive.

[–] siriusmart 5 points 6 months ago (1 children)

holdup taking the side length of the unit square actually makes the working out a bit simpler, because the otherside is 1

idk i just saw the diagonal at first

2l+0.5 sqrt(2l^2)=1
l(2+0.5sqrt(2))=1
l=1/(2+0.5sqrt(2))
=(2-0.5sqrt(2))/3.5
=(4-sqrt(2))/7

also, all the public items on the drive is mean to be public, there are unlisted and private items u dont see

[–] zkfcfbzr 3 points 6 months ago

Hey, you're right. I'm not sure what I did originally to make it so complicated, then ¯\_(ツ)_/¯