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## Rose Graph

https://www.desmos.com/calculator/outuspeasd

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## Cubes

You have a (solid) blue cube and an unlimited amount of (solid) red cubes, all of which are of the same size. What is the largest number of red cubes that can touch the blue cube along its sides or parts of its sides?

Touching along edges or at corner points do not count.

Problem comes from Pi Han Goh (https://brilliant.org/problems/he-was-wrong-too/?group=3BX9rjPrQ3CM&ref_id=820890)

Comment what you think is the answer, with some graphics if possible.

My 21 configuration that worked

Michael Mendrin’s 22 configuration.

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## Kissing Circles

So suppose you have 3 points in 2D space.

For each of these 3 points, a circle with it’s center at these points appears.

These 3 circles are not any other circles. They are tangent to each other, or they are kissing each other.

So given any 3 points, how do you find the radii of these 3 circles? Give a thought to this question. Solutions will come out tomorrow.

Solution:

Lets name the points ${p}_{1}$${p}_{2}$ and ${p}_{3}$, and the radii of each circle be ${r}_{1}$, , ${r}_{2}$ and ${r}_{3}$. Also, let the distance between ${p}_{a}$ and ${p}_{b}$ be ${D}_{a,b}$.

So now,

${D}_{1,2} = {r}_{1} + {r}_{2} \quad ---(1)$

${D}_{3,2} = {r}_{3} + {r}_{2} \quad ---(2)$

${D}_{1,3} = {r}_{3} + {r}_{1} \quad ---(3)$

From $(1)$, $(2)$ and $(3)$ we can find that:

${r}_{1}=\frac{{D}_{1,2}+{D}_{1,3}-{D}_{3,2}}{2}$

${r}_{2}=\frac{{D}_{1,2}+{D}_{3,2}-{D}_{1,3}}{2}$

${r}_{3}=\frac{{D}_{1,3}+{D}_{3,2}-{D}_{1,2}}{2}$

Hence solved.

https://www.desmos.com/calculator/jjm17kozmi