Answers to Puzzles

The Nine Dots

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o  o  o
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Weighing Coins

Divide the coins into three equal groups, A, B, and C, each with four coins.
Weigh A against B. There are two cases:

  • A and B balance: The counterfiet coin is in group C. Weigh two of the group C coins against two coins you know are good (from group A or B). The result of this weighing narrows the bad coin down to one of two. Weighing one of those two against a good coin completes the solution in this case.
  • A and B don't balance: Assume the heavy group is group A. In each pan, place two coins from group A and one coin from group B, leaving two group B coins aside. If the pans balance, the bad coin must be one of the two group B coins not on the scales, and we handle as above.
    If the pans don't balance, we've narrowed down to three coins: Either one of the two group A coins on the heavy side, or the single group B coin on the light side is the counterfiet. Weigh the two possibly bad A coins against each other. If they are equal, the bad coin is the B coin, and if they are unequal, the heavy one is the counterfeit.
For more info on this problem, see rec.puzzles:logic:weighing/balance.

The Dollar Hotel

Each person paid $9, totalling $27. The manager has $25 and the bellboy $2. The bellboy's $2 should be added to the manager's $25 or subtracted from the tenants' $27, not added to the tenants' $27.

Planning an Orchard

There are multiple answers, but the most elegant is to draw a five pointed star, and plant trees at the five points and five crossings.

The Math Department

17 weeks later, the entire faculty resigned at the luncheon. To understand this, imagine there are only two faculty. The week after the announcement, each assumes the other will resign. When that doesn't happen, it can only be because both professors have made errors, so they both resign the week after.

For a more detailed writeup of an identical problem, see rec.puzzles:logic:josephine or :smullyan/priest.

Tea and Cream

Exactly the same amount. First, a teaspoon of cream is added to the tea. Some amount x of cream goes back to the creamer in the second step, and (one teaspoon - x) of tea goes with it. Then the teacup contains (one teaspoon - x) of cream and the creamer contains (one teaspoon - x) of tea.

Bugs On A Square

Since the bugs start out walking perpendicularly, and there is nothing in the problem to alter this symmetry, the bugs are always walking perpendicularly. Since each bug is walking perpendicularly to the line separating it from the bug chasing it, the gap is closing at the speed of the chasing bug. Therefore, each bug walks a distance equal to the side of the square before it meets the next bug.

See rec.puzzles:analysis:bugs for more details.

River Crossing

Follow along with some objects:

  • Two cannibals cross, one comes back.
  • Two cannibals cross, one comes back.
  • Two missionaries cross, a cannibal and a missionary come back.
  • Two missionaries cross, a cannibal comes back.
  • Two cannibals cross.

More Crawling Bugs

Yes. Each time Igor stretches the rubber band, the bug comes with it, which means that the fraction of band left to cross stays the same.
After one crawl, the bug's done 1/3 of the band. Igor stretches the band to six inches (and the bug is now two inches along). The bug crawls one inch, which is now 1/6 of the total band. His next crawl will only be 1/9 of the band.
Then the fraction of band that the bug has crawled at n steps is:

1/3 + 1/6 + 1/9 + 1/12 + ... + 1/3n

or 1/3 (1 + 1/2 + 1/3 + 1/4 + ... + 1/n)

The sum in the middle is the first n terms of the harmonic series, which diverges. That is, if n is big enough the sum is as large as we want. In this case, 11 steps is enough.

The Unexpected Visit

This problem is paradoxical, and very difficult to explain. I refer to rec.puzzles:logic:unexpected for a partial answer. They in turn refer to the mathematical literature on the subject.

Connect The Dots

Yes. Of all the possible ways to connect up the dots in pairs, choose the way which has the shortest total length. This will not have any crossings. If it did have a crossing, we could replace the lines with shorter ones as in the pictures below, contradicting our choice of shortest total length.

R      B                      R------B
 \    /
  \  /
   \/      gets replaced by               which is shorter.
  /  \
 /    \
R      B                      R------B

All Triangles Are Isoceles

Unless the triangle is isoceles, the point O is outside the triangle. Given this, everything else in the proof is valid.

Stealing Rope

Almost all of it. Tie the ropes together. Climb up one of them. Tie a loop in it as close as possible to the ceiling. Cut it below the loop. Run the rope through the loop and tie it to your waist. Climb the other rope (this may involve some swinging action). Pull the rope going through the loop tight and cut the other rope as close as possible to the ceiling. You will swing down on the rope through the loop. Lower yourself to the ground by letting out rope. Pull the rope through the loop. You will have nearly all the rope.

A Fork In The Road

Ask: "If I were to ask you if the left road leads to Someplaceorother, would you say yes?"

The truthteller will say yes if the left road goes to Someplaceorother, no if not. The liar must answer the same.. the phrasing makes him lie twice and the two lies negate each other.

The Camel Race

The wise man tells them to switch camels.

The Bear

White. The only way the photographer could have taken his walk is near the north or south pole. Purists quibble about polar bear lifestyles, but why spoil an excellent problem.

Another Orchard

Four, if he has a good hill. One tree goes at the top of the hill, and three make a triangle at the hill's base. The four trees sit at the vertices of a tetrahedron.

Painting The Plane

This problem is unsolved. It's not hard to color the plane with seven colors, using hexagon tiles. It's not hard to show it can't be done with three colors. Nobody knows if 4,5 or 6 colors are possible.

The Cornerless Chessboard

No. Each domino covers one black square and one white square. Removing two opposite corners leaves an unequal number of white and black squares.

There Was A Frog...

28 days. On the 28th day, he crawls up three feet, leaves the well, and doesn't slip back the two feet at night.

Lamps And Switches

The most elegant solution is to turn on a switch and wait five minutes. Then turn on another switch and quickly enter the room. One bulb is off, one bulb is on, and one bulb is on and warm.

Let's Make A Deal

Yes. One third of the time, you picked the car originally and switching is bad. Two thirds of the time, you picked a goat originally, and switching will get you the car.

This problem becomes subtle if you worry about Monty's intentions. See rec.puzzles:decision:monty.hall for a more complete answer.

Wiring Problem

In fact, she can sort out any number of wires after only one trip to the roof and back. On the floor, connect the battery to one wire, and wire the others together in pairs. Climb the stairs. One wire will be hot, and that must be the one connected to the battery. Call that wire 1.
Pick another wire, label it 2, and connect it to wire 1. Wire 2 is connected to some other wire downstairs, so a third wire upstairs will go hot. Call it wire 3. Pick another wire, label it 4, and repeat. When this process is complete, the wires are connected like:

1  2   3  4   5  6   7  8   9 10   11
____   ____   ____   ____   ____ 
|  |   |  |   |  |   |  |   |  |    |
|  |   |  |   |  |   |  |   |  |    |
|  |   |  |   |  |   |  |   |  |    |
|  |   |  |   |  |   |  |   |  |    |
B  -----  -----  -----  -----  ------
although the electrician does not know the labels of the bottom wires. However, she goes back downstairs and cuts all connections except the battery. Wire 2 will be hot, which tells her wire 3 as well. Then reconnecting 2 to 3 turns 4 hot, and so on.

Get Rich With Gold Foil

In the second diagram, the "diagonal" of the rectangle is not right. Actually rearranging the pieces will leave a very thin parallelogram gap along the diagonal, with area exactly one square inch.

St. Ives

One. "I" was going to St. Ives. Sorry, it's a classic.

What Word Does This Represent?

M is under ST and in G. Misunderstanding.