# How many clues are there?

I have three teeny children; actually, their ages are 1, 3 and 6 (triangular numbers!). As such, excitement in our house peaks on birthdays, Christmas and Easter.

We recently had an Easter Egg hut and I hid 16 clues around our house. Each clue was just a single piece of paper: on one side was a number and on the other side was a letter.

Each of the 16 clues was uniquely numbered from 1 to 16, the idea being that when my kids find them all they can put them in order and then turn them over to see the message “BEHIND THE BOOKS”. This message is 16 characters long (including the two spaces). At this point, they can go and nudge the bookshelf forward to find a ridiculously large stash of chocolate hidden behind it.

After some searching, the children had collected all of the numbers from 1 to 16 but they were still searching for more numbers instead of getting on with the next bit.

And this brought to mind a small problem that I enjoyed solving whilst they kept looking.

Problem 1. If you are searching for clues that are uniquely numbered and you find yourself with the numbers from $1$ to $k$, what is the probability that there are any other clues left to find?

In the minds of my children, there could have been 20 clues and they just happened to have found the first 16 whilst leaving the numbers 17, 18, 19 and 20 unfound by chance.

The probability of this is quite small however, as you will see by solving the general form of the problem in Problem 1.

Another problem came to mind as well.

Problem 2. Let’s say that I have hidden $N$ clues and suppose that my children will stop searching as soon as they have found a complete sequence of clues from $1$ to $k$. What is the probability that they stop searching early?

## One comment

1. Excellent problem. After a fair amount of toiling, I realised this is a special case of the German tank problem, with the maximum serial number observed equal to the number of tanks. It notoriously gives different answers based on whether you hold a frequentist or bayesian philosophy, as well as the assumed prior distribution of the number of tanks/clues. Which explains the headache I induced in trying to even formally define the problem. Perhaps this early training will turn your children into exceptional WWII-era military generals!

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