## Tank counting

This week’s Riddler Classic is a simple-sounding problem about statistics. But it’s not so simple! Here is a paraphrased version of the problem:

There are $n$ tanks, labeled $\{1,2,\dots,n\}$. We are presented with a sample of $k$ tanks, chosen uniformly at random, and we observe that the smallest label among the sample is $22$ and the largest label is $114$. Both $n$ and $k$ are unknown. What is our best estimate of $n$?

Here is a short introduction on parameter estimation, for the curious:
[Show Solution]

Here is my solution to the problem:
[Show Solution]

## Rug quality control

This Riddler puzzle is about rug manufacturing. How likely are we to avoid defects if manufacture the rugs randomly?

A manufacturer, Riddler Rugs™, produces a random-pattern rug by sewing 1-inch-square pieces of fabric together. The final rugs are 100 inches by 100 inches, and the 1-inch pieces come in three colors: midnight green, silver, and white. The machine randomly picks a 1-inch fabric color for each piece of a rug. Because the manufacturer wants the rugs to look random, it rejects any rug that has a 4-by-4 block of squares that are all the same color. (Its customers don’t have a great sense of the law of large numbers, or of large rugs, for that matter.)

What percentage of rugs would we expect Riddler Rugs™ to reject? How many colors should it use in the rug if it wants to manufacture a million rugs without rejecting any of them?

Here is my solution:
[Show Solution]

## Finding the doctored coin

This Riddler puzzle is about repeatedly flipping coins!

On the table in front of you are two coins. They look and feel identical, but you know one of them has been doctored. The fair coin comes up heads half the time while the doctored coin comes up heads 60 percent of the time. How many flips — you must flip both coins at once, one with each hand — would you need to give yourself a 95 percent chance of correctly identifying the doctored coin?

Extra credit: What if, instead of 60 percent, the doctored coin came up heads some P percent of the time? How does that affect the speed with which you can correctly detect it?

Here is my solution.
[Show Solution]