Brain Teasers 25 - Puzzle Prime

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Heaven or Hell

Section: Brain Teasers Category: Insight

Difficulty: Advanced Tags: Featured

After you die, you somehow appear in a mystical room with two doors and two keepers inside. One of the doors leads to Heaven and the other door leads to Hell. One of the keepers is always lying and the other keeper is always saying the truth. If you can ask one of the keepers whatever question you want (you don’t know which keeper is lying and which one is truthful), how can you find your way to Heaven?

SOLUTION

You can point your finger to one of the two rooms and ask any of the keepers the question “If I ask the other keeper whether this room leads to Heaven, would he say YES?”. If the answer is NO, go through that door, if the answer is YES, go through the other one.

Unknown Author March 19, 2021

10 Comments

Camping Challenge

Section: Brain Teasers Category: Detective

Difficulty: Advanced

Look carefully at the picture below and answer the questions.

1. How many tourists are staying at this camp?
2. When did they arrive: today or a few days ago?
3. How did they get here?
4. How far away is the closest village?
5. Where does the wind blow: from the north or from the south?
6. What time of day is it?
7. Where did Alex go?
8. Who was on duty yesterday? (Give their name)
9. What day is it today?

SOLUTION

1. There are 4 people.
2. They arrived a few days ago, enough so that a spider web can appear on the tent.
3. Judging by the paddles, they got there with boats.
4. There is a hen walking around the camp, so the closest village is not far away.
5. The leaves of the trees are larger at the south side, so the wind must be blowing from the South.
6. The shadow is pointing towards West, so it must be morning.
7. Alex went to catch butterflies.
8. Since Peter is on duty today – cooking food for the group, it was Colin on duty yesterday.
9. Today is August 8. Watermelons ripen in August.

Unknown Author March 16, 2021

7 Comments

Knight Takes Rook

Section: Brain Teasers Category: Chess

Difficulty: Expert

White starts with e2-e4 and the game finishes with Knight takes Rook mate in the fifth move. Recreate the game.

SOLUTION

The game proceeds as follows:

1.e2-e4 Ng8-f6
2.Qd1-e2 Nf6xe4
3.f2-f3 Nd4-g3
4.Qe2xe7 Qd8xe7
5.Ke1-f2 Ng3xh1#

Unknown Author March 13, 2021

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Crashing Light Bulbs

Section: Brain Teasers Category: Insight

Difficulty: Hard

You are living in a 100-floor apartment block. You know that there is one floor in the block, such that if you drop a light bulb from there or anywhere higher, it will crash upon hitting the ground. If you drop a light bulb from any floor underneath it however, the light bulb will remain intact. If you have two light bulbs at your disposal, how many drop attempts do you need such that you can surely find which the floor in question is?

SOLUTION

The answer is 14 drops. You can do this by throwing the first bulb from floors 14, 27, 39, 50, 60, 69, 77, 84, 90, 95, 99, 100 (notice that the difference decreases always by 1) until it crashes and then start throwing the second bulb from the floors in between. For example, if the first bulb crashes at floor 69, you start throwing the second bulb from floors 61, 62, 63, etc. This way the total number of throws would be always at most 14.

Proving that 14 is optimal is done using the same logic. In order to use at most 13 throws, the first throw should be made from floor 13 or lower. The second throw should be made from floor 13+12 or lower, the third throw should be made from floor 13+12+11 or lower, etc. Continuing with the same argument, we conclude that the 13th drop should be made from floor 13+12+…+2+1=91 or lower. However, if the first light bulb does not crash after the last throw, you will not be able to find out which number among 92-100 is X.

Unknown Author March 7, 2021

8 Comments

9 balls, 1 defective

Section: Brain Teasers Category: Insight

Difficulty: Advanced Tags: Weighings

You have 9 balls, 8 of which have the same weight. The remaining one is defective and heavier than the rest. You can use a balance scale to compare weights in order to find which is the defective ball. How many measurements do you need so that you will be surely able to do it? What if you have 2000 balls?

SOLUTION

First, we put 3 balls on the left side and 3 balls on the right side of the balance scale. If the scale tips to one side, then the defective ball is there. If not, the defective ball is among the remaining 3 balls. Once left with 3 balls only, we put one on each side of the scale. If the scale tips to one side, the defective ball is there. If not, the defective ball is the last remaining one. Clearly we can not find the defective ball with just one measurement, so the answer is 2.

If you had 2000 balls, then you would need 7 measurements. In general, if you have N balls, you would need to make at least log₃(N) tests to find the defective ball. The strategy is the same: keep splitting the group of remaining balls into 3 (as) equal (as possible) subgroups, discarding 2 of these subgroups after a measurement. To see that you need no less than log₃(N) tries, notice that initially there are N possibilities for the defective ball and every measurement can yield 3 outcomes. If every time you get the worst outcome, you will make at least log₃(N) tries.

Unknown Author March 1, 2021

2 Comments

Coins on a Chessboard

Section: Brain Teasers Category: Mathematics

Difficulty: Hard

There is a room with a chessboard inside. On each of its 64 squares, there is placed a coin, either heads up or heads down. You enter the room and a person inside points towards one special square on the chessboard and gives you the chance to flip one of the coins (whichever you choose). Then you leave the room, your friend enters and has to guess which was the special square on the chessboard. If you two could devise a plan before entering the room, how would you make sure your friend always guesses correctly which is the special square?

SOLUTION

First, you must enumerate the coins with numbers from 1 to 64, locate the mystery coin, and calculate the binary representation of its number, padded with zeros on the left to 6 digits length. For example, if the mystery coin is the 5th one on the 4th row, its number would be 29 and will have a binary representation 011101. Then, consider the following sets of coins:

A1 = {row 1, row 2, row 3, row 4}
A2 = {row 1, row 2, row 5, row 6}
A3 = {row 1, row 3, row 5, row 7}
A4 = {column 1, column 2, column 3, column 4}
A5 = {column 1, column 2, column 5, column 6}
A6 = {column 1, column 3, column 5, column 7}

Now, the strategy is to flip the coin which makes the parity of heads in set Ai odd if and only if the i-th digit in the binary representation of the mystery coin is 1. It is easy to check that this is always a possible thing to do.

Unknown Author February 27, 2021

4 Comments

Drown or Burn

Section: Brain Teasers Category: Insight

Difficulty: Advanced

The ship you are traveling on crashes and you somehow succeed to reach the shores of an island. On this island however, a cruel tribe resides and decides to murder you. The tribals can not agree on how to do this, so they decide that if the first sentence you say is a lie, they will drown you, and if it is a truth, they will burn you. Luckily, you hear their conversation and come up with a plan. What do you tell them?

SOLUTION

You can tell them “You will drown me”. This will result in a paradox – if your sentence is true, then they will drown you, but on the other hand will be forced to burn you, which is a contradiction. Similarly if your sentence if false. Therefore they will not have a choice except to give up on their plan.

Unknown Author February 9, 2021

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The Majority Name

Section: Brain Teasers Category: Mathematics

Difficulty: Hard Tags: Algorithm

In a long list of names, one of the names appears more than half of the time. You will be read the names one at a time, without knowing how many they are, and without being able to write them down. If you have a very weak memory, how can you figure out which is the majority name?

SOLUTION

Remember the first name and then keep track of whether it has been repeated more than half of the time. To do that, simply add 1 if you hear the name or subtract one when you hear another name. If the list finishes and your counter is positive, then the first name is the majority. If your counter drops to 0, simply restart the procedure with the next name you hear.

This algorithm, invented by R. Boyer and J. Moore, works, because if the counter ends up at 0, then each of the names up to that moment has been read at most half of the time. Therefore, the majority name appears more than half of the time in the remainder of the list.

Unknown Author February 6, 2021

1 Comment

Under a Microscope

Section: Brain Teasers Category: Science

Difficulty: Easy

How much will a 12° angle measure when looked at under a microscope that magnifies 8 times?

SOLUTION

The angle will measure the same, 12°.

Unknown Author February 3, 2021

0 Comments

Ambiguous Clock

Section: Brain Teasers Category: Mathematics

Difficulty: Hard

The hands of my alarm clock are indistinguishable. How many times throughout the day their positioning is such that one cannot figure out which is the hour hand, and which is the minute hand?

Remark: AM-PM is not important.

SOLUTION

Imagine that you have a third hand which moves 12 times as fast as the minute hand. Then, at any time, if the hour hand moves to the location of the minute hand, the minute hand will move to the location of the imaginary hand. Therefore, our task is to find the number of times during the day when the hour hand and the imaginary hand are on top of each other, and the minute hand is not.

Since the imaginary hand moves 144 times faster than the hour hand, the two hands are on top of each other exactly 143 times between 12AM and 12PM. Out of these 143 times, 11 times all three arrows are on top of each other. Therefore, we have 2 × (143 – 11) = 264 times when we cannot figure out the exact time during the entire 24-hour cycle.

Unknown Author January 27, 2021

1 Comment