**Logic!**

A 4 * 4 * h rectangular box contains a sphere of radius 2 and eight smaller spheres of radius 1. The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is h?

Here’s an image

https://latex.artofproblemsolving.com/9/d/8/9d894dbf42883508d51bf4973c78d1368570c6fa.png

*The Last one (Answer to this is now at the bottom of my profile):*

10 people are sitting next to each other in a line. They all stand up, and either move 1 seat or stay where they are (people on the ends can’t move to the other end). Including the starting position, how many different ways can the people be seated after they move?

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Lebossle officially nominated scariest person ever, not only for solving this in about 15 seconds, but also for knowing literally everything every relating to math. If you want to feel inadequate, talk to him.

Anyways, for you normal people I will write a solution.

Let’s say the number of arrangements for 1 person is S1, 2 people is S2, 3 people is S3, etc. It’s easy to see S1=1 and S2=2. To find S3, consider the placement of the person on the edge.

Case A: human does not move

In Case A, there will be S2 possibilities, since the guy is essentially excluded.

Case B: human moves 1.

In Case B, there will only be S1 possibilities, since both the guy originally on the edge and the guy he switched with are locked into their positions.

Therefore, S3 = S2 + S1.

By the same line of reasoning, S4 = S3 + S2; S5 = S4 + S3, etc.

So, just find the 9th unique number of the Fibonacci sequence, namely, 89.