• ## Shorter Distance

Point D lies inside an arbitrary triangle ABC. Prove that |AB|+|BC|>|AD|+|DC|.

Zbigniew Michalewicz and David B. Fogel found this problem in a math text for fifth graders in the United States and gave it to many people, including undergraduate and graduate students and even full professors in mathematics, engineering, or computer science.

Fewer than 5% of those people solved it within an hour.

• ## Not-second, First, Second

Two players are playing the following game:

• Player A picks a sequence of three consecutive coin toss results, e.g., Tails-Heads-Heads, and tells it to player B
• Player B picks another sequence of three consecutive coin toss results, e.g., Heads-Tails-Heads, and tells it to player A
• A coin is flipped repeatedly and results are written down until one of the sequences is observed
• The player who’s sequence is observed first wins

Is the game fair?

• ## Box of Batteries

Is there enough space in the box to fit one more battery?

There are four cards on a table. Each card has a number on one side and a letter on the other. The visible faces of the cards show W, V, 42, and 7.

Which card or cards do you need to turn over to check whether the following statement is true:

“If a card has a V on one face, then the number on the opposite face is greater than 20

• ## Wheel Alignment

Are front wheels of a turning car parallel?

• ## Dividing with Straightedge

Given a line segment and a line parallel to it, divide the segment in three equal parts using only a straightedge (a ruler with no markings). You can draw straight lines, but not measure distances.

• ## The Diagonal

White to move.

• ## What's Wrong with this Picture?

What’s wrong with this picture?

• ## Look and Say Polynomial

1, 11, 21, 1211, 111221, 312211, 13112221, …

And the next element of the sequence is…

• ## Which Bicycle Wheel Turns Faster

Which bicycle wheel makes more turns over a given path — front or rear? Assume both wheels have the same diameter.

• ## The Last Knight

White to move.

• ## Risk and Dunbar Number

If you personally know somebody who died from X, then X is dangerous.

This risk assessment heuristic is not as stupid as it sounds.

• ## Your Friends Have More Friends than You Do

Henry   Heather
|   |
Helen Cindy Carlos Olivia Harold
|   |
Harry   Hector
• ## Two Envelopes

You are given two envelopes. Each envelope contains some money. You can pick one envelope, count the money inside it, then either keep this envelope, or return it and keep the other one.

Can you do better than just picking an envelope at random and keeping it?

• ## Stick+Pencil

When would somebody hold the stick like shown, and why?

• ## Where to Wait for an Elevator

You are waiting for an elevator. There are four elevators, equally spaced along a wall. The third elevator is out of order. Where would you stand, and why?

• ## Determining Cat Chirality

Does your cat prefer to sleep like this ⟳ (clockwise), like this ⟲ (counterclockwise), or has no preference? Let’s figure it out:

• Print this chart
• Put a pin or a dot in the lower left corner (at coordinates (0,0))
• When you observe ⟳, move the pin right by one
• When you observe ⟲, move the pin up by one
• If the pin reaches the red line, stop the test. Cat prefers ⟲
• If the pin reaches the green line, stop the test. Cat prefers ⟳
• If the pin reaches the yellow line, stop the test. No preference detected
• ## Raymond Smullyan, 1957

The white king has just been knocked off the board. Where was it standing?

• ## Everyday Carry: Car

Contents of the small backpack I keep in the trunk and take with me when hiking.

• ## Carlsen-Karjakin 2016, Game 16

White to move.

• ## Is 333,333,333,333,333,331 Prime?

 31 is prime. 331 is prime. 3,331 is prime. 33,331 is prime. 333,331 is prime. 3,333,331 is prime. 33,333,331 is prime.

Is 333,333,333,333,333,331 prime?

• ## Dragon Curve

Take a strip of paper and fold it in half to the right. Then fold it in half again, in the same direction. Repeat several times, then open out so that each fold becomes a 90 degree turn. This example uses paracord instead of paper, but idea is the same.

• ## A Sin to Err

A toy pen and paper cipher

• ## Right Triangle Area

Find the area of the right triangle with hypotenuse 10 and height (dropped from the right angle) 6.

This problem was on a math test for quite some time, until it turned out that the best students (and only the best students) could not solve it.

• ## Irrational Exponentiation

What is greater, $e^{\pi}$ or $\pi^e$?

• ## Four Moves

This is the position after black move 4. What were the moves?

• ## Rolleron

Back when electronic components were heavy and expensive, this nifty contraption was used to prevent air-to-air missiles from spinning en route to their target.

• ## Switch from macOS to Linux

(E)migration from macOS to Linux is complete.

Left: early 2013 13 inch MacBook Pro Retina with 2.6 GHz Intel Core i5 running macOS 10.12.4.

Right: early 2017 Dell XPS 13 with 2.7 GHz Intel Core i7-7500 running Ubuntu 16.04.

• ## Square from Four Points

You are given one point from each side of a square. Reconstruct the square with compass and straightedge.

• ## Richard Réti, 1921

White to move.

• ## Not Quite Random

Let’s say we want to estimate $\pi$ with Monte Carlo. We get some pairs of random numbers $(x, y)$ with both $x$ and $y$ uniformly distributed between 0 and 1. Approximately $\frac{\pi}{4}$ of those should fall within the unit circle, so we can get an estimate of $\pi$ by multiplying proportion of points that fall within that circle by 4.

Knowing that some implementations of pseudorandom number generators are better than others, we decide to take a look at the first 256 points. Can you tell which of the two pictures below shows uniformly distributed random numbers?

• ## Carlsen-Karjakin 2016, Game 8

Black to move.

• ## Comparing Logs

What is greater, $log_2 3$ or $log_3 5$?

• ## Aerial Photography

An airplane is taking pictures of the ground. Its camera is facing straight down. The camera is loaded with a long roll of film that is slowly advanced by a motor.

Come up with a system that controls the rate of advance so that the film moves at the same speed as the image projected by the lens, regardless of airplane speed and altitude.

Complication: only use technology available in 1950’s. Photosensors and basic electronics is ok, but no CCD and no computers.

• ## Leonid Kubbel, 1922

White to move.

• ## Estimating BDBA Probability

DBA (Design Basis Accident) is an accident that a nuclear facility is designed to withstand.

BDBA (Beyond Design Basis Accident) is an accident that was not fully considered in the design process because it was judged to be too unlikely.

Let’s try to quantify how unlikely.

• ## Richard Réti, 1922

White to move.

• ## Calculating 52! by Hand

It’s easy - calculate 54!, then divide by 53 and 54.

• ## Under Construction

In 2017 I plan to publish here one blog post per week, starting on January 1st.