Can you hop to the lily pad?

This week’s Fiddler is about hopping back and forth.

You are a frog in a pond with an infinite number of lily pads in a line, marked “1,” “2,” “3,” etc. You are currently on pad 2, and your goal is to make it to pad 1. From any given pad, there are specific probabilities that you’ll jump to another pad: Whenever you are on pad $k$, you will hop to pad $k−1$ with probability $1/k$, and you will hop to pad $k+1$ with probability $(k−1)/k$.

What is the probability that you will ultimately make it to pad 1?

My solution:
[Show Solution]

2025 puzzle

This week’s Fiddler is about the number 2025, in celebration of (almost) New Years!

First puzzle: What is the greatest number of distinct primes that add up to 2025?

Second puzzle: How can you assign a set of 20 distinct prime numbers to the 20 vertices of a dodecahedron, so that the numbers on the five vertices of each face add up to 2025?

My solution:
[Show Solution]

Particles in a box

This week’s Fiddler is an optimization problem about fitting particles in a box.

You have three particles inside a unit square that all repel one another. The energy between each pair of particles is $1/r$, where $r$ is the distance between them. To be clear, the particles can be anywhere inside the square or on its perimeter. The total energy of the system is the sum of the pairwise energies among the particles. What is the minimum energy for $9$ particles, and what arrangement of the particles produces it?

My solution:
[Show Solution]

Halloween Puzzle

This week’s Fiddler is about rounding!

You are presented with a bag of treats, which contains $n \geq 3$ peanut butter cups and some unknown quantity of candy corn kernels (with any amount being equally likely). You reach into the bag $k$ times, with $3 \leq k \leq n$, and pull out a candy at random. Each time, it’s a peanut butter cup! How many candy kernels do you expect to be in the bag?

My solution:
[Show Solution]

Round, round, get a round

This week’s Fiddler is about rounding!

Let $\text{round}(x)$ be the value of $x$ rounded to the nearest integer. Suppose $x_1,\dots,x_n$ are independent uniformly distributed random variables in $[0,1]$. Find the probability that
\[
\text{round}(x_1+\cdots+x_n) = \text{round}(x_1)+\cdots+\text{round}(x_n)
\]

My solution:
[Show Solution]

Tiling a Tilted Square

This week’s Fiddler is a challenging counting problem.

Consider the following array of 25 squares:

You are filling the array with rectangles by repeating the following two steps:

  1. Select one of the 12 squares along the outer perimeter that has not yet been selected as part of a rectangle.
  2. Form the largest rectangle you can that includes the square you just selected and other squares that are not yet part of any such rectangle.

You repeat these steps until every square along the perimeter has been selected. Here are two final states you might encounter:

How many distinct final states are possible? (Note: States that are rotations or reflections of each other should be counted as distinct.)

My solution:
[Show Solution]

How many times can you add up the digits?

This week’s Fiddler is a puzzle about adding digits over and over again.

For any positive, base-10 integer $n$, define $f(n)$ as the number of times you have to add up its digits until you get a one-digit number. For example, $f(23) = 1$ because $2+3 = 5$, a one-digit number. Meanwhile, $f(888) = 2$, since $8+8+8 = 24$, a two-digit number, and then adding up those digits gives you $2+4 = 6$, a one-digit number. Find the smallest whole number $n$ such that $f(n) = 4$.

Extra Credit: For how many whole numbers $n$ between $1$ and $10,000$ does $f(n) = 3$?

My solution:
[Show Solution]

Chessboard race

This week’s Fiddler is a puzzle with a surprise connection to physics!

A tiny ant is racing across a 2-by-2 chessboard, as shown below, where each smaller square has a side length of 1 cm. The ant starts at the bottom-left corner of the bottom-left black square and is trying to reach the top-right corner of the top-right black square. The ant moves faster on the white squares than on the black squares. Speed on the white squares is 1 cm per minute, while speed on the black squares is 0.9 cm per minute. What’s the least amount of time it will take the ant to reach the finish?

Extra Credit: Instead, the board is now 8-by-8, as shown below.

My solution:
[Show Solution]

Pancake race

This week’s Fiddler is a logic puzzle about getting home as fast as possible.

Alice, Bob, and Carey start together and each walk home at a different constant speed. Once all three get home, they can have pancakes! Alice can walk home in 10 minutes, Bob can do it in 20, and Carey in 30. Fortunately, any of them can carry any of the others on their back without reducing their own walking speed. Assume that they can pick someone up, set someone down, and change direction instantaneously. What is the fastest they can get to eat pancakes?

Extra Credit
There is now a fourth: Dee. Dee is the slowest, needing 60 minutes to walk home. As before, anyone can carry anyone else, and they won’t get pancakes until everyone gets home. What is the fastest this can happen?

My solution:
[Show Solution]

Dungeon Master’s Dice

This week’s Fiddler is a probability question about a dice-rolling game.

Two people are sitting at a table together, each with their own bag of six “DnD dice”: a d4, a d6, a d8, a d10, a d12, and a d20. Here, “dX” refers to a die with X faces, numbered from 1 to X, each with an equally likely probability of being rolled. Both people randomly pick one die from their respective bags and then roll them at the same time. For example, suppose the two dice selected are a d4 and a d12. The players roll them, and let’s further suppose that both rolls come up as 3. What luck! What’s the probability of something like this happening? That is, what is the probability that both players roll the same number, whether or not they happened to pick the same kind of die?

Extra Credit
Instead of two people sitting at the table, now suppose there are three. Again, all three randomly pick one die from their respective bags and roll them at the same time. For example, suppose the three dice selected are a d4, a d20, and a d12. The players roll them, and let’s further suppose that the d4 comes out as 4, the d20 comes out as 13, and the d12 comes out as 4. In this case, there are two distinct numbers (4 and 13) among the three rolls. On average, how many distinct numbers would you expect to see among the three rolls?

My solution:
[Show Solution]