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Heaven or Hell

Section: Brain Teasers Category: Insight
Difficulty: Tags:

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?

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.

puzzle
Unknown Author March 19, 2021
10 Comments

Camping Challenge

Section: Brain Teasers Category: Detective
Difficulty:

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?

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.

puzzle
Unknown Author March 16, 2021
7 Comments

9 balls, 1 defective

Section: Brain Teasers Category: Insight
Difficulty: Tags:

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?

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.

puzzle
Unknown Author March 1, 2021
2 Comments

Drown or Burn

Section: Brain Teasers Category: Insight
Difficulty:

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?

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.

puzzle
Unknown Author February 9, 2021
0 Comments

Ten Lanterns

Section: Brain Teasers Category: Deduction
Difficulty:

You have ten lanterns, five of which are working, and five of which are broken. You are allowed to choose any two lanterns and make a test that tells you whether there is a broken lantern among them or not. How many tests do you need until you find a lantern you know for sure is working?

Remark: If the test detects that there are broken lanterns, it does not tell you which ones and how many (one or two) they are.

You need 6 tests:

(1, 2) → (3, 4) → (5, 6) → (7, 8) → (7, 9) → (8, 9)

If at least one of these tests is positive, then you have found two working lanterns.

It all of these tests are negative, then lantern #10 must be working. Indeed, since at least one lantern in each of the pairs (1, 2), (3, 4), (5, 6) is not working. Therefore, there are at least 2 working lanterns among #7, #8, #9, #10. If #10 is not working, then at least one of the pairs (7, 8), (7, 9), or (8, 9) must yield a positive test, which is a contradiction.

puzzle
Unknown Author December 27, 2020
2 Comments

Diagonal in a Rectangle

Section: Brain Teasers Category: Mathematics
Difficulty:

A 1000 × 1004 rectangle is split into 1 × 1 squares. How many of these squares does the main diagonal of the large rectangle pass through?

Notice that the number of small squares the main diagonal passes through is equal to the number of horizontal and vertical lines it intersects. Indeed, every time the diagonal goes through the interior of one square to the interior of another, it must intersect one of these lines.

There are 1000 + 1004 = 2004 lines which are intersected by the main diagonal. However, on four occasions (which is the greatest common divisor of 1000 and 1004), the main diagonal intersects one horizontal and one vertical line at the same time, which results in double-counting., so we must subtract 4 from the answer.

Therefore, the answer is 1000 + 1004 – 4 = 2000.

puzzle
Unknown Author December 18, 2020
0 Comments

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