Discovery in the '100-Day Challenge'

In Day 82:

According to this picture by @David Vreken , we know that if the size of the table is $a\times b$ , then the total number of collisions is:

$(\frac{{\rm lcm}(a,b)}{a}-1)+(\frac{{\rm lcm}(a,b)}{b}-1)=\frac{a+b}{\gcd{(a,b)}}-2$

In Day 83:

According to my method, if the cup capacity is $a$ and $b$ respectively, and the cup is empty at the beginning, then the total plan that can be poured is

$\frac{a+b}{\gcd{(a,b)}}$

These two formulas have similarities, what is the relationship between them?

Easy Math Editor

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`2 \times 3`	$2 \times 3$
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Comments

Same diagram

A Former Brilliant Member - 9 months, 3 weeks ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$