Touch or Don’t Touch

For this puzzle/game, you need to keep presenting various words to your friends and telling them whether they are “TOUCH” or they are “DON’T TOUCH”. Below, we have listed several words from each category.

TOUCH: banana, proof, mouse, keyboard, promo, woman

DON’T TOUCH: cherry, solution, cat, screen, discount, girl

Can you guess what determines whether a word is “TOUCH” or “DON’T TOUCH”?

Words that make your lips touch when pronounced belong to the “TOUCH” category, while the others belong to the “DON’T TOUCH” category. The sounds “P”, “B”, “M”, and “W” that cause this are called “bilabial”.

Cut the Pizza

Cut a circular pizza into 12 congruent slices, such that exactly half of them contain crust.

Remark: We say that a slice contains crust if it shares an arc with the boundary of the pizza (with non-zero measure).

First, cut the pizza into 6 congruent circular triangles, and then split each of them in half, as shown on the image below.

Spot the Robber

The streets of the city are a square grid that extends infinitely in all directions. One of the streets has a police officer stationed every 100 blocks and there is a robber is somewhere in the city.

Can you devise a strategy that guarantees the robber will be spotted by a police officer at some point, no matter how he tries to avoid them?

Note: The officers can see infinitely far, but their running speeds are lower than the speed of the robber.

Let the police officers are located at points with coordinates (100N, 0) for N = 0, ±1, ±2… First, we fix the positions of all officers stationed at points (±200N, 0), then repeatedly perform the following procedure, step by step:

On step M, we let the non-fixed officers who are closest to the center move to the free points with coordinates (K, 0) and (0, K) for K = 0, ±1, ±2, … ±M. Then we fix their positions.

Since there are fixed officers at points (200N, 0) at all times, the robber is contained within some vertical strip the entire time. Therefore, at some point there will be two fixed officers that will restrict the robber within a horizontal segment of size 1, at coordinates (x, T) for x (S, S+1) and some T. Finally, at some point an officer will move to the point (0, T) and will spot the robber.

Clean Death

A man was going to bleach his socks because they had gotten muddy the day before. As he was pouring the bleach into the washing machine, he spilled some on the floor. He got some cleaning fluid and mopped it up with a rag. Minutes later he was dead. What killed him?

When the ammonia (NH3), found in cleaning fluids, is mixed with bleach (a dilute solution of sodium hypochlorite, NaOCl), a deadly gas (monochloramine, NH2Cl) is produced that can kill a person instantly. The resulting chemical reactions are the following:

  1. Bleach decomposes, forming hydrochloric acid and subsequently chlorine gas:
    NaOCl → NaOH + HOCl
    HOCl → HCl + O
    NaOCl + 2HClCl2 + NaCl + H2O
  2. The chlorine gas reacts with the sodium hypochlorite in the bleach, releasing chloramine as a vapor:
    2NH3 + Cl2 → 2NH2Cl

The Bicycle Problem

If you pull straight back on a pedal of a bicycle when it is at its lowest position, will the bicycle move forward or backward?

The surprising answer is that (usually) the bicycle will move backward.

When a bicycle moves forward, the trajectory its pedal traces with respect to the ground is called a trochoid. Depending on the selected gear of the bicycle, that trochoid could be:

  1. Curtate trochoid (for almost all gears of most bicycles)
  2. Prolate trochoid (if the gear is very low and the bicycle moves slowly)
  3. Common trochoid, a.k.a. cycloid (if the wheels of the bicycle and the pedal spin at identical speeds, practically never happens)
curtate trochoid
prolate trochoid
common trochoid (cycloid)

Since we are fixed with respect to the ground, by pulling the pedal backward, we are causing it to move leftward along the trochoid and therefore the bicycle will be moving backward. We note that despite that, the pedal will be moving forward with respect to the bicycle (but not with respect to the ground).

You can see a visual explanation of this puzzle in the video below.

Leave No Squares

How many matchsticks do you need to remove so that no squares of any size remain?

Nine matchsticks are enough, as seen from the solution below.

To see that eight matchsticks are not enough, notice that removing an inner matchstick reduces the number of 1×1 squares at most by 2. Since there are 16 such small squares, in order to get rid of them all, we need to remove only inner matchsticks. However, in this case, the large 4×4 square will remain.