# Critical Problem Solving

The best way to become better at
solving critical problems is to practice, practice, practice. Challenge yourself to
figure out the answer before **clicking the title of the entry for the answer.**

## Cube Walking

**Question:**

What is the shortest distance to get from one corner of a unit cube to the extreme opposite corner, without leaving the surface of the cube?

## Strategic Counting Game

**Question:**

You and Player A are playing a competitive game, taking turns calling out integers. The first person to call out “50” wins. Here are the rules:

- The player who start must call out an integer between 1 and 10, inclusive.
- On each successive turn, a new number called out must exceed the most recent number called by at least 1 and by no more than 10. For example, following 9, anything from 10 to 19, inclusive, can be called out.

A) To win regardless of what the other player does, shouldyou want to go first, and what would be the strategy in the game?

B) What if the rule is altered so that the first person tocall out “100” wins?

C) What about any integer n > 10 to win the game? Wouldyou still want to start first in this game?

## To Switch or Not to Switch

**Question:**

You are given choice of four envelops. Inside one envelop is $240; inside the others, fake bills. You pick one envelop but do not open it. Of the three remaining envelopes, one of the empty envelops is opened and revealed. You are offered the choice to pick one of the other two untouched envelops. However, you are told that if you switch your pick and subsequently end up with envelop with the real bill, you only keep a fraction, p, of the amount in there.

A) What is the value of p that will make you indifferent about switching your initial pick or not?

B) Finally, in a similar scenario but with n number of envelops, what is p? In other words, what is the function p(n)?

**Tags:**

## Successive Remainders

**Question:**

What is the smallest positive number that leaves a remainderof 1 when divided by 2, a remainder of 2 when divided by 3 … and a remainder of 9 when divided by 10?

## Who Is Wearing Which Hat

**Question:**

A dark closet has three blue and two red hats. Three men with that information go into the closet, randomly choose a hat, and come out. They each can’t see the hat on their own head. The first man looks at the other two and claims that he can’t tell what color his hat is. The second man hears this, looks at the other two, and says the same. The third man is blind, but says that he knows what the color of his hat is. What color is it?

## Queue of Prisoners with Hats

**Question:**

There are 100 prisoners and an officer. The officer gave the command to the prisoner that next day they will be given a hat to wear -- either red or blue. All of the prisoner will be standing in a line, and will not be allowed to communicate in any forms to each other. Each prisoner can’t see the hat on his own head, but can see the hats of everyone in front of him. The officer will start asking the color of the prisoner's hat one by one from the back, and immediately reply whether the response was correct or not. If everyone can identify the color of his own hat, with one mistake allowed overall, the entire group will be freed. Any further mistakes, the entire group will not be freed. What strategy should the prisoners enact for tomorrow?

## Two Traps

**Question:**

Without any wrongdoings, you are brought into an interrogation room by the governor and told that “You can make one last statement to save yourself before we seize one item of yours. We will seize your house if you tell a lie; we will seize your car if you say a truth.” You don’t want to lose either your house or car. What one statement can you say to save yourself?

## Three Islanders

**Question:**

There are two newly discovered islands. The Islanders born on one island always tell the truth, while the Islanders from the other island always lie. You are on one of the islands, and meet three Islanders. You ask the first Islander which island they are from, and he indicates that the other two Islanders are from the same Island he is from. You ask the second Islander the same question, and he also indicates that the other two Islanders are from the same island he is from. Can you guess what the third Islander will answer to the same question? If so, what’s the response?

## Number of Lies

**Question:**

You are presented with fourstatements in a list:

- A. The number of false statements in the list is zero.
- B. The number of false statements in the list is one.
- C. The number of false statements in the list is two.
- D. The number of false statements in the list is three.
- E. The number of false statements in the list is four.

You are also presented astatement that’s not on the list: F. All five statements on the list above are false statements. Out of the options A through F, which of the above statements has to be true?

## Pot of Beans

**Question:**

A pot contains 75 white beans and 150 green ones. Next to the pot is a large pile of green beans. A cook removes two beans from the pot at random. If at least one of the beans is green, he places it on the bean-pile and drops the other bean back into the pot, no matter what color, back in the pot. If both beans are white, on the other hand, he discards both of them and removes one green bean from the pile and drops it in the pot. At each turn of this procedure, the pot has one less bean in it. Eventually, just one bean is left in the pot. What color is it?

## Handshakes

**Question:**

At a networking event, everyone shook hands with everyone else. In total, there were 78 handshakes.

A) How many people werethere?

B) Suppose this same group of people met a week later for another networking event. Everyone again shook hands with everyone else, except that one of the people at the event refused to shake hands with anyone, claiming that he already knew everyone. How many handshakes were at this second event?

## Togglers

**Question:**

There are 5 people in front of you. One of them is the truthteller and always says the truth. The other four are togglers: they maytell the truth or lie the first time you ask him/her a question. However, if the toggler told the truth the first time, the next time he/she will lie. Reversely, if the toggler lied the first time, he/she will tell the truth the second time. How can you determine who the truthteller is, with only 2 questions that may be addressed to anyone?

## Pizza Galore

**Question:**

How many square feet of pizza are eaten in the United States each month? Is the answer closest to:

a) Area of New Jersey?

b) Area of a typical classroom?

c) Area of Central Park?

d) Area of a baseball stadium?

e) Area of available office space in the new World Trade Center?

**Tags:**

## Boys and Girls

**Question:**

At the beginning of year t=1, a city contains 1,000 couples but no children. Suppose each family wishes to have a daughter and has one baby per year until the arrival of the first girl. Assume children are equally liked to be born boy or girl. Let p be the expected percentage of children in the city that are boys at the end of the year t. How does p(t) change over time?

## Dice Game

**Question:**

You roll a single die no more than three times. You receive in payment the number of dollars as there are dots on the single face on the last roll of dice. What is the expected payoff of this game?

**Tags:**

## Randomly Choosing Socks

**Question:**

You have 6 red socks, 4 green socks and 2 blue socks in a basket.

A) What is the minimum number of socks you have to pick to make sure you have at least a pair of socks that don’t match in color?

B) Suppose that your answer from Part A is n. What’s the probability that you needed the n-th pick in order to get a mismatch pair?

**Tags:**

## Climbing Snail

**Question:**

A snail is climbing up a 101-foot pole. It climbs up by three feet everyday during the “working hours,” but each night while it sleeps, it slides down by one foot. When exactly does it reach the top of the pole?

## Mathematical Steps

**Question:**

You start at the number 2. On each step, you can multiply your current number by 2, divide your current number by 2, or square the current number.

A) How many steps would it be necessary to get to the number 8?

B) What about 32?

C) What about 2048, which is 2^11?

D) What about 32768, which is 2^15?

E) What about 2^24?

## Penny in Bottle

**Question:**

There's a penny in a bottle and a cork at its mouth. You can't pull the cork out or break the bottle. How do you get the penny out?

## Squares on Chessboard

**Question:**

How many total squares are there on a standard chessboard? A chessboard is made up of unit squares for an 8x8 total grid. Hint: the answer is not 64.