JamshrivelsonPuzzle solutions for you.
http://joshmaxsilverman.github.io
The radius of a rectangle
What would the world look like if you could only see squares? -
Question: One of my favorite facts about circles is the relationship between their area and their circumference. For a circle with radius $r,$ its area is $\pi r^2$ and its circumference is $2\pi r.$ What’s neat here (or rather, one thing that’s neat here) is that if you take the...
Sun, 19 May 2024 00:00:00 -0700
http://joshmaxsilverman.github.io/2024-05-19-differential-radius/
http://joshmaxsilverman.github.io/2024-05-19-differential-radius/Can you beat the heats?
A good old racetrack whodunit. -
Question: There are $25$ sprinters at a meet, and there is a well-defined order from fastest to slowest. That is, the fastest sprinter will always beat the second-fastest, who will in turn always beat the third-fastest, and so on. However, this order is not known to you in advance. To...
Mon, 06 May 2024 00:00:00 -0700
http://joshmaxsilverman.github.io/2024-05-06-time-trials-detective/
http://joshmaxsilverman.github.io/2024-05-06-time-trials-detective/Capture the flag
How often can you win when you have a number and they have an arrow? -
Question: It’s been a while and change, but the Robot Games are back once again. This time it’s Capture the Flag! Two robots, Aaron and Erin, have made it to this year’s final! Initially they are situated at the center of a unit circle. A flag is placed somewhere inside...
Wed, 01 May 2024 00:00:00 -0700
http://joshmaxsilverman.github.io/2024-05-01-capture-the-flag/
http://joshmaxsilverman.github.io/2024-05-01-capture-the-flag/Card collisions
What's the chance no card sees its own reflection? -
Question: You and your friend both have standard decks of $52$ cards. First, you combine them into a single deck with $104$ cards, and you thoroughly shuffle it. Then, you randomly split this back into two decks with $52$ cards each—one for you, and one for your friend. Each of...
Sun, 21 Apr 2024 00:00:00 -0700
http://joshmaxsilverman.github.io/2024-04-21-cooperative-card-game/
http://joshmaxsilverman.github.io/2024-04-21-cooperative-card-game/Paint by number
Cluster your expectations -
Question: Once again, I’m painting an infinitely long strip of canvas, broken up into adjacent $1$ cm-by-$1$ cm squares. Squares are randomly and independently numbered $0$ or $1.$ But this time, the strip itself is $2$ cm wide. Squares are considered adjacent if they share a common edge. So squares...
Tue, 09 Apr 2024 00:00:00 -0700
http://joshmaxsilverman.github.io/2024-04-09-cluster-size/
http://joshmaxsilverman.github.io/2024-04-09-cluster-size/Pick a circle, any circle
What chance is there, for a circle inside a square? -
Question: A circle is randomly generated by sampling two points uniformly and independently from the interior of a square and using these points to determine its diameter. What is the probability that the circle has a part of it that is off the square? (Jane Street) Solution There are two...
Mon, 19 Feb 2024 00:00:00 -0800
http://joshmaxsilverman.github.io/2024-02-19-pick-a-circle/
http://joshmaxsilverman.github.io/2024-02-19-pick-a-circle/Traffic light chase
How long until you meet your menacing assailant? -
Question: You find yourself engaged in a car chase, but it differs from the movies in one key way: Both you and the car that’s chasing you obey all traffic laws, being sure to stop at any and all red lights. You’re trying to make it to the city limit,...
Mon, 23 Oct 2023 00:00:00 -0700
http://joshmaxsilverman.github.io/2023-10-23-traffic-light-chase/
http://joshmaxsilverman.github.io/2023-10-23-traffic-light-chase/Pinball lights
How many shots does it take to turn the lights on? -
Question: You’re playing a game of pinball that includes four lanes, each of which is initially unlit. Every time you flip the pinball, it passes through exactly one of the four lanes (chosen at random) and toggles that lane’s state. So if that lane is unlit, it becomes lit after...
Sun, 01 Oct 2023 00:00:00 -0700
http://joshmaxsilverman.github.io/2023-10-01-pinball-lights/
http://joshmaxsilverman.github.io/2023-10-01-pinball-lights/Unbalanced league
How often will geography rob your team of a playoff berth? -
Question: Among all six divisions in Fiddler League Baseball (with five teams per division), what is the probability that there exist two divisions such that every team in one division has a higher winning percentage than every team in another division? (Note that this includes cases where multiple divisions are...
Mon, 18 Sep 2023 00:00:00 -0700
http://joshmaxsilverman.github.io/2023-09-18-unbalanced-league/
http://joshmaxsilverman.github.io/2023-09-18-unbalanced-league/Cool weave curve
What shape is the margin doodle? -
Question: A weaving loom set comes with a square with equally spaced hooks along each of its sides, as well as elastic bands that can be attached to the hooks. Suppose a particular weaving loom has $N$ hooks on each side, evenly spaced from one corner to another (i.e., there...
Sun, 10 Sep 2023 00:00:00 -0700
http://joshmaxsilverman.github.io/2023-09-10-cool-weave-drawing/
http://joshmaxsilverman.github.io/2023-09-10-cool-weave-drawing/Buffon's prism
If you throw a rod into a jelly lattice, what is the chance it crosses just one wall? -
Question: Consider 3-space (i.e. $\mathbb{R}^3$) partitioned into a grid of unit cubes with faces defined by the planes of all points with at least one integer coordinate. For a fixed positive real number $\ell$, a random line segment of length $\ell$ (chosen uniformly in location and orientation) is placed in...
Tue, 08 Aug 2023 00:00:00 -0700
http://joshmaxsilverman.github.io/2023-08-08-buffons-prism/
http://joshmaxsilverman.github.io/2023-08-08-buffons-prism/Twisted balkanization
How many different adventures does the Mobius prism have in store? -
Question: Instead of a strip of paper, consider a three-dimensional prism whose bases are regular $N$-gons. I twist it and stretch it into a loop, before finally connecting the two bases. Suppose that my twist is by a random angle, such that the two bases are aligned when they are...
Sun, 09 Jul 2023 00:00:00 -0700
http://joshmaxsilverman.github.io/2023-07-09-mobius-prism/
http://joshmaxsilverman.github.io/2023-07-09-mobius-prism/A chancy race
Will the coiners flip before you roll? -
Question: With The Riddler nearing its end here at FiveThirtyEight, I can finally get something off my chest: Starting a competition with the flip of a coin (say, to determine possession of a ball) is so boring! Instead, let’s give the captain of one team a fair coin and the...
Sun, 18 Jun 2023 00:00:00 -0700
http://joshmaxsilverman.github.io/2023-06-18-die-coin-538/
http://joshmaxsilverman.github.io/2023-06-18-die-coin-538/Take a number
When should you stop dreaming of a brighter future? -
Question: You start with just the number 1 written on a slip of paper in a hat. Initially, there are no other slips of paper in the hat. You will draw from the hat $100$ times, and each time you draw, you have a choice: If the number on the...
Mon, 22 May 2023 00:00:00 -0700
http://joshmaxsilverman.github.io/2023-05-22-drawing-game/
http://joshmaxsilverman.github.io/2023-05-22-drawing-game/Save the world
Good use of polling or bad use of polling? -
Question: Later this year, aliens will visit Earth and announce that they intend to blow up the planet. (Lovely.) However, they present you with a challenge. If you successfully complete the challenge, they’ll blow up another planet instead (probably Neptune, because why not). The aliens have telepathically assigned each of...
Mon, 15 May 2023 00:00:00 -0700
http://joshmaxsilverman.github.io/2023-05-15-save-the-world-538/
http://joshmaxsilverman.github.io/2023-05-15-save-the-world-538/Hard choices for the Easter bunny
Hoppity hop, which carton are you going to get? -
Question: For Easter, you and your family decide to decorate $10$ beautiful eggs. You pull a fresh carton of eggs out of your fridge and remove $10$ eggs. There are two eggs remaining in the carton, which you return to the fridge. The next day, you open the carton again...
Sun, 09 Apr 2023 00:00:00 -0700
http://joshmaxsilverman.github.io/2023-04-09-easter-bunny/
http://joshmaxsilverman.github.io/2023-04-09-easter-bunny/Left handed sister
Are you talking to some brother, or the brother? -
Question: Suppose you meet someone who looks like the brother of your friend Mary. You ask if he has a sister named Mary, and he says “Yes I do, but I don’t think I know you.” You remember that Mary has a sister who is left-handed, but you don’t remember...
Thu, 06 Apr 2023 00:00:00 -0700
http://joshmaxsilverman.github.io/2023-04-06-left-handed-sister/
http://joshmaxsilverman.github.io/2023-04-06-left-handed-sister/Robot long jump
Jump late, and long, but not too late. -
Question: Great news! The variety of robotic competition continues to grow at breakneck pace! Most recently, head-to-head long jump contests have been all the rage. These contests consist of rounds in which each robot has a single attempt to score. In an attempt, a robot speeds down the running track...
Sat, 01 Apr 2023 00:00:00 -0700
http://joshmaxsilverman.github.io/2023-04-01-robot-long-jump/
http://joshmaxsilverman.github.io/2023-04-01-robot-long-jump/NCAA quota bracket
I'll take one of each. -
Question: It feels like there’s more parity in college basketball’s March Madness than ever, with lower-seeded teams advancing further in the tournaments at the expense of the favorites. This year’s Sweet $16$ on the men’s side consists of two $1$ seeds, two $2$ seeds, two $3$ seeds, two $4$ seeds,...
Sun, 26 Mar 2023 00:00:00 -0700
http://joshmaxsilverman.github.io/2023-03-26-ncaa-quota-bracket/
http://joshmaxsilverman.github.io/2023-03-26-ncaa-quota-bracket/Fast times at the marble track
It won't be long before this marble's at your door -
Question: While passing the time at home one evening, you decide to set up a marble race course. No Teflon is spared, resulting in a track that is effectively frictionless. The start and end of the track are $1$ meter apart, and both positions are $10$ centimeters off the floor....
Sun, 12 Mar 2023 00:00:00 -0800
http://joshmaxsilverman.github.io/2023-03-12-fast-marbles/
http://joshmaxsilverman.github.io/2023-03-12-fast-marbles/