10 Dots, 10 Coins

If you have 10 dots on the ground, can you always cover them with 10 pennies without the coins overlapping?

Assume the dots lie in a plane and the radius of a penny is 1. Make an infinite grid of circles with radii 1, as shown on the picture, and place it randomly in the plane.

If we choose any point in the plane, the probability that it will end up inside some circle of the grid is equal to S(C)/S(H), where S(C) is the area of a coin and S(H) is the area of a regular hexagon circumscribed around it. A simple calculation shows that this ratio is larger than 90%. Therefore, the probability that some chosen point in the plane will not end up inside any circle is less than 10%. If we have 10 points, the probability that neither of them will end up inside a circle is less than 100%. Therefore, we can place the grid in the plane in such a way that every dot ends up in some circle. Now, just place the given coins where these circles are.

Married Couples

In a small village, there are 100 married couples living. Everyone in the village lives by the following two rules:

  1. If a husband cheats on his wife and she figures it out, the husband gets immediately killed.
  2. The wives gossip about all the infidelities in town, with the only exception that no woman is told whether her husband has cheated on her.

One day a traveler comes to the village and finds out that every man has cheated at least once on his wife. When he leaves, without being specific, he announces in front of everybody that at least one infidelity has occurred. What will happen in the next 100 days in the village?

Let us first see what will happen if there are N married couples in the village and K husbands have cheated, where K=1 or 2.

If K = 1, then on the first day the cheating husband would get killed and nobody else will die. If K = 2, then on the first day nobody will get killed. During the second day, however, both women would think like this: “If my husband didn’t cheat on me, then the other woman would have immediately realized that she is being cheated on and would have killed her husband on the first day. This did not happen and therefore my husband has cheated on me.”. Then both men will get killed on the second day.

Now assume that if there are N couples on the island and K husbands have cheated, then all K cheaters will get killed on day K. Let us examine what will happen if there are N + 1 couples on the island and L husbands have cheated.

Every woman would think like this: “If I assume that my husband didn’t cheat on me, then the behavior of the remaining N couples will not be influenced by my family’s presence on the island.”. Therefore she has to wait and see when and how many men will get killed in the village. After L days pass however and nobody gets killed, every woman who has been cheated on will realize that her assumption is wrong and will kill her husband on the next day. Therefore if there are N + 1 couples on the island, again all L cheating husbands will get killed on day L.

Applying this inductive logic consecutively for 3 couples, 4 couples, 5 couples, etc., we see that when there are 100 married couples on the island, all men will get killed on day 100.

The Father

Mary is 21 years older than her son. After 6 years, she will be 5 times older than him. Where is the father? 

Let M be the age of the mother and S be the age of the son. We have M = S + 21 and M + 6 = 5(S + 6). We solve the system and get S= -3/4, i.e. minus 9 months. Therefore right now the son just got conceived and the father is with the mother.

Normal Person

I caused my mother’s death and didn’t get convicted.
I married 100 women and never got divorced.
I got born before my father, but I am considered perfectly normal?

Who am I?

A priest, whose mother dies from labor, who marries 100 women to 100 men, and whose father attends his birth.