# Puzzles

Things I've done:

# Chestnuts

This is a collection of puzzles from a class I taught in the mid 1990's. These puzzles are 'chestnuts', which means they:

• Are short and easy to tell people
• Drive you nuts thinking about them
• Have answers that make you kick yourself
• Have been around forever

If you're interested in more classic puzzles, try the rec.puzzles archive.

Finally, there are no answers to these puzzles at this site. If you're really stuck on a specific problem, write me at bryan@slu.edu and I'll help you out.

## The Nine Dots

Draw four straight lines that hit all nine dots in the picture below, where each line must start where the last one ended. (You can't lift your pencil off the page).

```o  o  o

o  o  o

o  o  o
```

## Weighing Coins

Suppose you have twelve gold coins, all of which look exactly the same. Unfortunately one of them is fake, and not made of gold. It doesn't weigh the same as the others, but you don't know if it is heavier or lighter. To find the coin, you have a two-pan balance. How can you find the fake coin using only three weighings?

(By a 'weighing', I mean you put some coins in each pan and see which way the balance tilts.)

## The Dollar Hotel

Three guys get a hotel room for the night, and the concierge charges them \$30 for it, or \$10 each. They go up to their room, and the concierge realizes that the room they were getting was only a \$25 room, so he sends the bellhop up with \$5 for the guys.

Well, the bellhop decides the three guys will have trouble splitting the \$5 up amongst themselves, so he helpfully pockets \$2 and returns \$1 to each guy.

The problem is this - each guy paid \$9 for the room, and the bellhop has \$2, for a total of \$29. What happened to the extra dollar?

## Planning an Orchard

A farmer with a big flat farm wants to plant ten trees to make five rows, with four trees in each row. How can this be done?

(Obviously, some trees will need to be in more than one row)

## The Math Department

Podunk University once boasted 17 tenured professors of mathematics. Tradition prescribed that at their weekly luncheon, faithfully attended by all 17, any members who had discovered an error in their published work should make an announcement of this fact, and promptly resign. Such an announcement had never actually been made, because no professor was aware of any errors in her or his work. This is not to say that no errors existed, however. In fact, over the years, in the work of every member of the department at least one error had been found, by some other member of the department. This error had been mentioned to all other members of the department, but the actual author of the error had been kept ignorant of the fact, to forstall any resignations.

One fateful year, the department was augmented by a visitor from another university, Prof. Smith, who had come with hopes of being offered a permanent position at the end of the academic year. Naturally, he was appraised, by various members of the department, of the published errors which had been discovered. When the hoped-for appointment failed to materialize, Prof. Smith obtained his revenge at the last luncheon of the year. "I have enjoyed my visit here very much," he said, "but I feel that there is one thing that I have to tell you. At least one of you has published an incorrect result, which has been discovered by others in the department." What happened in the next year?

## Tea and Cream

A man is pouring tea for his daughter, and after filling the cup, he takes a teaspoon of cream (from an identical cup) and stirs it into her tea. She wails "no daddy! I didn't want cream", takes the spoon and puts a spoonful of tea back into the cream cup.

Is there more tea in the cream, or more cream in the tea?

## Bugs On A Square

Four bugs are placed at the corners of a square. Each bug walks always directly toward the next bug in the clockwise direction. How far do the bugs walk before they meet?

## River Crossing

Three missionaries and three cannibals are traversing the jungle, and they come to a river. They fashion a crude raft, but it only holds two. The missionaries know that if there are ever more cannibals than missionaries on one bank, the cannibals will gang up and eat the missionaries.

How do they cross the river?

## More Crawling Bugs

A bug is crawling along one of Igor's really high quality rubber bands, which has been cut, and is now a three inch long strip of rubber. Every time the bug crawls an inch along the strip, Igor cruelly stretches it another three inches.

Will the poor bug ever finish?

## The Unexpected Visit

One Friday afternoon, Bob's boss desends from the heights and tells Bob: "I plan to visit your operation one day next week to make sure you're still useful. And I promise you you won't expect me when you come in to work on the day I visit."

Bob frets all weekend, but then realizes something. The boss can't visit on Friday, because if Bob came to work on Friday he would certainly expect his boss. But if the boss can't visit on Friday, then if Bob makes it to Thursday he'll expect his boss on Thursday, so the boss can't visit on Thursday.

Bob rules out Wednesday, too. See, the boss can't visit on Thursday or Friday, so if Bob makes it to Wednesday, he'll expect the boss. And the boss is supposed to be unexpected. Similar logic rules out Tuesday, and then Monday.

So, Bob decides his boss can't visit at all, and takes it easy. Unfortunately for Bob, the boss shows up unexpectedly on Wednesday, and Bob is fired.

So what was wrong with Bob's reasoning?

## Connect The Dots

Scatter an equal number of red and blue dots on a sheet of paper. You can connect each red dot to a different blue dot with a straight line, so that they all pair off. The question is: Can you always pair them off so that none of the lines cross?

## All Triangles Are Isoceles

The picture and description below it outline a proof that all triangles are isoceles (having two equal sides). The statement is blatantly false, but what is wrong with the proof?

Draw the bisector of angle ABC and the perpendicular bisector of side AC. These meet at a point O. Now construct OQ and OR perpendicular to AB and BC respectively. Finally, draw segments AO and OC.

Now, AP = PC, OP = OP, and angles APO and CPO are right. Thus triangles APO and CPO are congruent. Also, BQO and BRO are right angles, BO = BO, and angle QBO = RBO since BO is the bisector. Thus triangles QBO and RBO are congruent and we have QB = RB.

Finally, by the last paragraph, AO = CO and QO = RO. Since AQO and CRO are right angles, triangles AQO and CRO are congruent by Hypotenuse-Leg.

Now AQ = RC and QB = RB. Thus AB = CB and triangle ABC is isoceles.

## Stealing Rope

Two 50-foot ropes are hanging next to each other from a 50-foot high ceiling. Armed with a knife but nothing else, how much rope can a clever thief steal?

## A Fork In The Road

Two men stand at a fork in the road. One fork leads to Someplaceorother; the other fork leads to Nowheresville. One of these people always answers the truth to any yes/no question which is asked of him. The other always lies when asked any yes/no question. By asking one yes/no question, can you determine the road to Someplaceorother?

## The Camel Race

An Arab sheik tells his two sons to race their camels to a distant city to see who will inherit his fortune. The one whose camel is slower will win. The brothers, after wandering aimlessly for days, ask a wise man for advice. After hearing the advice they jump on the camels and race as fast as they can to the city. What did the wise man say?

## The Bear

A photographer goes bear hunting. From camp, he walks one mile due south, then one mile due east, then shoots a bear, and, after walking one more mile, is back at camp. What color was the bear?

## Another Orchard

A farmer wants to plant an orchard so that every tree is exactly 20 feet from every other tree in the orchard. How many trees can he plant?

## Painting The Plane

Suppose you want to paint the Euclidean plane so that any two points which are exactly one unit apart are different colors. How many colors do you need to use?

## The Cornerless Chessboard

It is easy to cover a typical 8x8 chessboard with dominos that cover two squares each. Now suppose you remove two diagonally opposite corner squares from the chessboard. Can you still cover what's left of the board with dominos?

## There Was A Frog...

There was a frog at the bottom of a thirty foot well. Each day, he crawled his way three feet up the side of the well. Then, each night, he slid two feet back down. How long did it take the frog to escape the well?

## Lamps And Switches

Outside of a closed room there are three on/off light switches, which control three lamps inside the room. You can't see inside the room while flipping switches, and you can't flip the switches from inside the room. How do you determine which switch controls which lamp while only entering the room once?

## Let's Make A Deal

You are a participant on "Let's Make a Deal." Monty Hall shows you three closed doors. He tells you that two of the closed doors have a goat behind them and that one of the doors has a new car behind it. You pick one door, but before you open it, Monty opens one of the two remaining doors and shows that it hides a goat. He then offers you a chance to switch doors with the remaining closed door. Is it to your advantage to do so?

## Wiring Problem

In the elevator shaft of a tall building, there are 11 identical electrical wires running from the top to the bottom. The electrician needs to figure out which wires on the bottom correspond to which wires up top. She's got one battery and a voltage tester, and she can connect and disconnect any number of wires at either end. How can she solve the problem with the fewest trips up and down the stairs?

## Get Rich With Gold Foil

Take an 8" by 8" sheet of gold foil, 64 inches square, and cut it up as in the picture on the left. Take the pieces and rearrange them to form the 5" by 13" rectangle on the right, giving you 65 square inches of gold foil. Now keep one square inch as profit, melt down the rest and repeat for an early retirement.

What's going on?

## St. Ives

As I was going to St. Ives,
I met a man with seven wives.
Each wife had seven sacks,
each sack had seven cats,
and each cat had seven kittens.
Kittens, cats, sacks, wives, how many were going to St. Ives?