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dailymaths·Daily Maths Challengesbyzkfcfbzr

Find a+b

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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Comments8
zkfcfbzr
lemmy.world

Since the meat of the problem is in the proof, I'll mention the value of a + b is:

::: spoiler a + b = 11 :::

​ ::: spoiler further commentary There are probably a few ways to prove this. I'll mention at least one of those ways is both simple and short, within the grasp of your typical sophomore in high school. :::

​

I don't plan to make this into a regular series like siriussmart's, but I might occasionally post problems I come across that seemed interesting.

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siriusmartreply
lemmy.world

hey would u like to create an index post? its just so that everything can be accessed within a few clicks from the listing post

u can follow the format

# Index of XXX

|Date|Post|Difficulty|
|---|---|---|
|2024/04/28| [Sum to infinity is -1/12](https://lemmy.world/post/14877972)|7/10|
...

its entirely optional and its up to u

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zkfcfbzrreply
lemmy.world

I've got 3 or 4 other problems in mind that I might post at some point - I'll make an index post with problem #2. Wanna keep them spaced out since I don't have too many.

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zkfcfbzrreply
lemmy.world

All good 👍

::: spoiler 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.

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siriusmartreply
lemmy.world

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

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zkfcfbzrreply
lemmy.world

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

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