When is a triangle like a circle?

This week’s Fiddler is about a generalized notion of “radius”.

For a circle with radius $r$, its area is $\pi r^2$ and its circumference is $2\pi r$. If you take the derivative of the area formula with respect to $r$, you get the circumference formula! Let’s define the term “differential radius.” The differential radius $r$ of a shape with area $A$ and perimeter $P$ (both functions of $r$) has the property that $dA/dr = P$. (Note that $A$ always scales with $r^2$ and $P$ always scales with $r$.)

For example, consider a square with side length $s$. Its differential radius is $r = s/2$. The square’s area is $s^2$, or $4r^2$, and its perimeter is $4s$, or $8r$. Sure enough, $dA/dr = d(4r^2)/dr = 8r = P$. What is the differential radius of an equilateral triangle with side length s?

Extra credit:
What is the differential radius of a rectangle with sides of length $a$ and $b$?

My solution:
[Show Solution]

Tiling squares

This week’s Fiddler is about tiling a square with smaller squares.

Suppose you have infinitely many 3-by-3 cm tiles and infinitely many 5-by-5 cm tiles. You want to use some of these tiles to precisely cover a square whose side length is a whole number of centimeters. Tiles may not overlap, and they must completely cover the larger square, without jutting beyond its borders. What is the smallest side length this larger square can have, such that it can be precisely covered using at least one 3-by-3 tile and at least one 5-by-5 tile?

Extra credit:
This time, you have an infinite supply of square tiles for each odd whole number side length (as measured in centimeters) greater than 1 cm. In other words, you have infinitely many 3-by-3 cm tiles, infinitely many 5-by-5 cm tiles, infinitely many 7-by-7 cm tiles, and so on. You want to use one or more of these tiles to precisely cover a square whose side length is $N$ cm, where $N$ is an integer. Once again, tiles may not overlap, and they must completely cover the larger square without jutting beyond its borders. What is the largest integer N for which this task is not possible?

My solution:
[Show Solution]

Showcase Showdown

This week’s Fiddler is based on “Showcase Showdown” on the game show “The Price is Right”.

Suppose we have some number of players. Player A is the first to spin a giant wheel, which spits out a real number chosen randomly and uniformly between 0 and 1. All spins are independent of each other. After spinning, A can either stick with the number they just got or spin the wheel one more time. If they spin again, their assigned number is the sum of the two spins, as long as that sum is less than or equal to 1. If the sum exceeds 1, A is immediately declared a loser.

After A is done spinning (whether once or twice), B steps up to the wheel. Like A, they can choose to spin once or twice. If they spin twice and the sum exceeds 1, they are similarly declared the loser. This continues until all players are done. Whoever has the greater value (that does not exceed 1) is declared the winner.

Assuming all players play the game optimally, what are player A’s chances of winning?

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]

Don’t flip out

This week’s Fiddler is a probability question about a coin-flipping game.

Kyle and Julien are playing a game in which they each toss their own fair coins. On each turn of the game, both players flip their own coin once. If, at any point, Kyle’s most recent three flips are Tails, Tails, and Heads (i.e., TTH), then he wins. If, at any point, Julien’s most recent three flips are Tails, Tails, and Tails (i.e, TTT), then he wins.

However, both players can’t win at the same time. If Kyle gets TTH at the same time Julien gets TTT, then no one wins, and they continue flipping. They don’t start over completely or erase their history, mind you—they merely continue flipping, so that one of them could conceivably win in the next flip or two.

What is the probability that Kyle wins this game?

Extra Credit
Kyle and Julien write down all eight possible sequences for three coin flips (HHH, HHT, HTH, THH, HTT, THT, TTH, and TTT) on eight different slips of paper. They place these slips into a hat and shake it.

They will each randomly draw slips of paper out of the hat, at which point they will play the same game as previously described, but looking for the sequence specified on the slip of paper they each selected. Kyle draws first and looks at his slip of paper. After doing some calculations, he says: “Well, at this point, it’s about as fair a match as it could possible be.”

Which slip or slips of paper might Kyle have drawn? And what are his chances of winning at this point (i.e., before Julien selects his own slip of paper)?

My solution:
[Show Solution]

The Likeliest Monopoly Square

This week’s Fiddler is about rolling dice in the board game Monopoly.

We have a square board with 40 individual spaces around it, numbered from 0 to 39. All players begin on space 0 (akin to the “Go” square in Monopoly) and roll a pair of dice to determine how many spaces they advance each turn. However, unlike Monopoly, there is no way to otherwise advance around the board (i.e., there’s no “Chance,” “Community Chest,” going to jail, etc.). In their first pass around the board, which space from 1 to 39 are players most likely to land on at some point (i.e., not necessarily on their first or last roll, but after any number of rolls)?

Extra Credit
The square board has 10 spaces on each side. The first side has spaces 0 through 9, the second side has spaces 10 through 19, the third side has spaces 20 through 29, and the fourth side has spaces 30 through 39. Because you’re rolling two dice, it’s impossible to land on space 1 in your first pass around the board. Several other spaces on the first side of the board are similarly unlikely. Putting that first side of the board aside, which space from 10 to 39 are players least likely to land on at some point during their first pass around the board? (Another question: What if you rolled three dice at a time instead of two?)

My solution:
[Show Solution]

Picking a speaker at random

This week’s Fiddler is about selecting a speaker of the house at random. How long will it take?

There are three candidates who want the job of Speaker. All 221 members of the party vote by picking randomly from among the candidates. If one candidate earns the majority of the votes, they become the next Speaker. Otherwise, the candidate with the fewest votes is eliminated and the process is repeated with one less candidate. If two or more candidates receive the same smallest number of votes, then exactly one of them is eliminated at random. What is the average number of rounds needed to select a new Speaker?

Extra Credit
What if there were 10 candidates running for Speaker?

My solution:
[Show Solution]