Saturday, June 17, 2017

The Monk and the Mountain Path

One morning at precisely 9:00 AM a monk begins walking up a mountain path. He takes his time, stopping several times to rest along the way. He arrives at the temple at the mountain's summit at precisely 5:00 PM that evening. The next day, the monk leaves the temple at precisely 9:00 AM and makes his way back down the path. Again, he takes his time and rests at several points along the journey. He arrives back at his original starting point at precisely 5:00 PM that evening. Is there any time when the monk is in exactly the same spot on both days? Click below to see the answer.

Saturday, June 10, 2017

The Pigeonhole Principle

The pigeonhole principle states that if a group of pigeons flies into a set of pigeonholes, and there are more pigeons than pigeonholes, then there must be at least one pigeonhole with two pigeons in it. More generally, if k + 1 or more objects are placed into k boxes, then there is at least one box containing two or more of the objects. Despite its seeming simplicity (perhaps obviousness), it can be used to solve a surprising range of problems in probability, number theory, and computer science, just to name a few. See if you can use it to solve the following three problems.

  1. (Warm up) A drawer contains a dozen blue socks and a dozen black socks, all unmatched. If the room is dark, how many socks do you have to take out to be sure you have a matching pair?
  2. Prove that there are at least two people in Tokyo with exactly the same number of hairs on their heads.
  3. Prove that if five distinct integers are selected from the numbers 1 through 8, there must be at least one pair with a sum equal to 9.

Click below to see the answers.

Saturday, June 3, 2017

Coffee with Cream

Suppose you have two cups in front of you, one with precisely 8 fluid ounces of coffee, and the other with precisely 8 fluid ounces of cream. You take precisely one teaspoon of the cream and add it to your coffee. You stir it in so that it's thoroughly mixed. Then you take precisely one teaspoon of that coffee/cream mixture and put it back into the cup of cream. Does the cup of coffee have more cream in it, or does the cup of cream contain more coffee? Click below for the answer.

Saturday, May 27, 2017

Letter Groupings

The letters of the (English) alphabet can be grouped into four distinct categories.





Based on the categories established by the first 13 letters, can you place each of the remaining 13 letters in the correct group?

Saturday, May 20, 2017

Number Words

In the solution to A Unique Number, I asked a bonus question. "Can you think of a number whose letters when spelled out in English are all in alphabetical order?" Several people replied via Twitter with the correct answer of "forty." You may have found a shortcut to the solution if you noted that none of the single-digit numbers have their letters in alphabetical order, nor does the word "teen." This allows you to skip ahead to 20, 30, etc. Can you use a similar strategy to answer the following questions?
  • What is the lowest number that requires the five vowels A, E, I, O, and U only once each in its spelling?
  • What is the lowest number that requires the six letters A, E, I, O, U, and Y only once each in its spelling?
Click below to see the answers.

Saturday, May 13, 2017

The Nine Dot Puzzle

The following is a classic "thinking outside the box" puzzle. Can you connect all nine dots below by drawing exactly four straight lines, without lifting your pencil or tracing back over any line?

Give it a try before you click below for the answer.

Saturday, May 6, 2017

Apples and Oranges

You work in a factory boxing fruit. In front of you are three boxes labeled "apples," "oranges," and "apples & oranges." One box contains only apples, one contains only oranges, and one contains a mixture of both apples and oranges. Unfortunately, the label machine has gone haywire and has mislabeled all three boxes. Can you look at one piece of fruit from only one of the boxes and correctly label all three? Click below for the solution.