# A Puzzle by Midnight Math

Imagine a game that involves betting on the color of a single card in a standard 52-card deck. Each card is turned over one by one, and before each card is flipped, you may do one of two things:

1) Bet: If the next card is red, you earn \$1. If the next card is black, you lose \$1.

2) Pass: The next card is turned over and shown to you, and play continues.

Once you bet on a card, the deck is reshuffled and may play again. If you reach the end of the deck, you are forced to bet on the last card.

In a naive strategy, you could bet on the first card of each deck, winning 50% of the time and earning, on average, \$0. Can you produce a better strategy, or a proof that one does not exist?

Send in your solutions (with proofs) to midnight.math@gmail.com or talk to Kevin O’Toole or Ian Hoover. If you are correct, you will be given the highest of accolades: your name mentioned here, next issue.

# A Puzzle By Midnight Math

Kevin buys a bus ticket from Singapore to Malaysia, but he is unaware that his ticket has an assigned seat. Upon boarding the bus, he sits in a random empty seat. Passengers continue to board and sit in their assigned seats until a passenger sees Kevin in their seat. The passenger, not wanting to be rude and displace Kevin, assumes they have made a mistake and sits in a random empty seat. The next passenger whose seat was taken also sits in a random seat, and so on until the last passenger sits in the last seat left. Given that the order of passengers entering (including Kevin) is random, and that each passenger (who isn’t Kevin) will sit in their assigned seat unless it’s taken, and that Kevin is not the last passenger, what is the probability that the last passenger in line will sit in their assigned seat?

Send in your solutions (with proofs) to midnight.math@outlook.com. If you are correct, you will be given the highest of accolades: your name mentioned here, next issue.

# A Puzzle by Midnight Math

New Puzzle
Try to imagine an irrational number between 0 and 1, represented in decimal, such that nowhere in the infinite sequence of digits does the same digit repeat twice in a row. Does such a number exist? If so, show why. If not, show why not.

# A Puzzle by Midnight Math

New Puzzle

A prince is surveying his kingdom. No matter where he stands, if he walks 100 paces forward in a straight line, turns to the left 90 degrees and walks 100 paces forward again in a straight line, again turns 90 degrees to the left and walks 100 more paces in a straight line, he arrives where he began walking. Where is he? (Bonus points, who is he?)

# A Puzzle By Midnight Math

You have a pizza, but your slicer is only good for 10 cuts. You want to serve as many people as possible before your slicer gives out. How many is this? What about if your pizza slicer was good for n cuts? Put another way, what is the maximum number of partitions a circle can be partitioned into by making n chords?”

Midnight Math is a club run by Ian Hoover ’15 who just returned from studying away in Spain.
Send in your solutions (with proofs) to midnight.math@outlook.com. If you are correct, you will be given the highest of accolades: your name mentioned here, next issue.

# A Puzzle by Midnight Math

Monge’s Theorem, stated informally, says that regardless of the size or location of the three circles depicted above, the points A, B, and C will form a straight line. Prove it.

Send your solutions to this problem (with proofs) to midnight.math@outlook.com. If you are correct, you will be given the highest of accolades: your name mentioned here, next issue.

# A Puzzle by Midnight Math: September 2013

Midnight Math is run by Kevin O’Toole ’15.
Have you ever wanted a slice of pizza with no crust? Do you usually feed your crust your dogs?

Find a way of cutting a circular pizza into finitely many congruent pieces such that at least one piece has no crust.

More formally, find a set of simply connected regions (X1, X2…Xn) such that:

• The intersection (X1 U X2 U… Xn) is the unit disk, D, on ℝ2.
• For each i, j < n there is a rigid, possibly orientation-reversing transformation of the plane which converts Xi to Xj
• For some i, λ(Xi ∩ ∂D) = 0, where λ is the Lebesgue measure*, and ∂D is the boundary of the unit disk, D.

# A Puzzle by Midnight Math: May

The passel of math students from the first ever Midnight Math puzzle again find themselves held captive by that rascally hyper-intelligent, pan-dimensional being.

This time they are each held in separate cells and each day, one is chosen at random to come to a special room to work on a generalized integral transform problem (each selection is completely independent of any of the previous choices).

# A Puzzle by Midnight Math: April

Imagine you have n points evenly spaced around a circle. Choose one of these as the starting point, p0. Take a pen and draw a line skipping m – 1 points so that you connect p0 to pm.

Continue doing this without lifting the pen (so next you connect pm to the point m points later, and so on). Eventually you get back where you started.

Perhaps you have drawn an n-pointed star (the classic way of drawing a 5 pointed start is with n = 5 and m = 2).

Why can you never get a six pointed start this way? What other values of n will never result in a star?

# A Puzzle by Midnight Math: March

Let’s examine the following procedure: Start with a finite string of digits and replace each substring consisting of a repeated single digit with the number of digits in that substring followed by the digit of that that is being repeated. Example: 333 would become 33, and 2 would become 12 and 222233333 would become 4253. Starting with 1, and recursively applying this procedure generates the sequence: 1, 11, 21, 1211,… and so on. What is the largest digit that will ever appear in this sequence?