April 2025:
I can choose to continually flip a fair coin until I see HHTT. Or I can choose
to flip the coin until I see HTHT. On average should it take more flips for HHTT or HTHT?
March 2025:
Lines are placed independently at random on the plane. Consequently
the plane is divided into many polygons. If you select one of the formed polygons at random. How many
sides would you expect it to have?
February 2025:
Given a circle of unit radius. What is the average distance of a point sampled within the unit circle to the center?
January 2025:
Lines are placed independently at random on the plane. Consequently
the plane is divided into many polygons. If you select one of the formed polygons at random.
How many sides would you expect it to have?
December 2024:
Find a simple way to establish the following identity:
\[
\lim_{n \rightarrow \infty} \int_0^1\int_0^1 \cdots \int_0^1 \frac{x_1^2 + x_2^2 + \cdots + x_n^2}{x_1 + x_2 + \cdots + x_n} dx_1 dx_2 \cdots dx_n = \frac{2}{3}
\]
November 2024:
Player A has 100 fair coins and player B has 101 fair coins. Both of them
toss their respective coins simultaneously and count the number of heads they each respectively received.
Whoever has more heads is declared the winner. What's the probability that player B beats player
A?
October 2024:
Given a 6 by 6 grid of points, can you remove 6 of the points so that there is an even number of points in each row, column, and the two main diagonals?
September 2024:
You place a large number of points randomly in the unit square, what is the distribution of the radius of the largest circle containing no points?
August 2024:
A quarter is glued to a tabletop. You then place another
quarter tangent to the tabletop quarter and begin rotating it around the tabletop quarter. You keep rotating
it until your quarter reaches the point it started at. How many times will your quarter rotate?
July 2024:
You have a coin that is biased in favor of heads. What can you do to turn it into a fair coin?
June 2024:
I have an equilateral triangle comprised of 9 unit equilateral triangles. I shade each unit triangle either red or blue with equal probability.
We say that a sub-triangle is comprised of 4 unit equilateral triangles. What's the probability that there will be no blue sub-triangle?
May 2024:
I have a convex pentagon with a total perimeter of size 1. I then line the floor with parallel
lines each spaced 1 unit apart from each other. I then toss the pentagon like a coin. What is
the probability that it intersects a line?
April 2024:
I have two coins. One is fair, the other is biased towards heads. You are allowed only two flips and then must guess which coin is biased.
What strategy increases your odds of determining which coin is biased?:
1. Pick one of the coins and then flip it twice.
2. Flip each coin once.
Pi Day 2024:
Download the puzzles
here. Enjoy!
March 2024:
The Game of Pig:
You start with a score of 0 and roll a fair 6 sided dice.
* If you roll a 2,3,4,5 or 6, add this number to your current score.
* If you roll a 1, you go bust and lose the game.
* You can choose to stop rolling at any time and keep your score.
A common strategy in this push-your-luck dice game is to choose a goal score `s`, where you
stop rolling if you get `s` or more points. What is the optimal choice of `s`?
February 2024:
I have assigned two uniformly distributed random numbers from -1000 to 1000 on two slips of paper and turn them face down.
You are allowed to turn the first slip over and examine the value. If you believe it's the larger of the two values, you keep
the slip of paper. If not, you keep the second slip of paper. You win if you get the larger of the two values.
Is it possible to come up with a strategy that let's you win more than 50% of the time?
January 2024:
What is the probability that three uniformly random points on a circle will be contained in some semicircle?
December 2023:
Take a circle and draw two chords at random on the circle. What's the probability that the two
chords intersect?
November 2023:
A man in a gallery is asked about a portrait depicting a man with an object obscuring his face.
He responds, 'Brothers and sisters I have none, but that man's father is my father's son.'
What is the object obscuring the face of the man in the portrait?
October 2023:
I have 12 identical pieces of candy. In how many ways can I distribute the candies to 4
different kids?
September 2023:
I flip a fair coin until I see two heads (HH) or until I see a heads followed by a tails (HT).
Should the number of expected flips until I see HH be less than, equal to, or larger than the expected
number of flips until I see a HT?
