The Temple of Doom

You and eight of your team members are trying to escape the Temple of Doom. You are running through a tunnel away from a deadly smoke and end up in a large hall. There are four paths ahead, and exactly one of them leads to the exit. It takes 20 minutes to explore any of the four paths one way, and your group has only 60 minutes until the deadly smoke suffocates you. The problem is that two of your friends are known to be delirious and it is possible that they do not tell the truth, but nobody knows which ones they are. How should you split the group and explore the tunnels, so that you have enough time to figure out which is the correct path and escape the temple?

You explore the first path. You send two of your teammates to explore the second path. You send the remaining six teammates in groups of three to explore each of the two remaining paths. If your path leads to the exit, then everything is good. Otherwise, you ask the two groups of three whether their paths lead to the exit. If in both groups everyone answers consistently, then nobody is lying, and you will escape. If in both groups there is a person whose answer is different from the others in the group, then the majority in both groups says the truth. Once again, you will know which path leads to the exit. Finally, if in exactly one of the groups everyone answers consistently, you ask the group of two. If the team members there answer consistently with each other, then they say the truth. You will have two groups which tell the truth and will know which path leads to the exit. If the answers of the teammates in the group of two differ, then in the inconsistent group of three the majority will be saying the truth. Again, you will be able to deduce which path leads to the exit.

Cucumbers

You had 10lbs of cucumbers, each of which consisted of 99% water. After leaving them in the sun, some of the water in the cucumbers evaporated. If the cucumbers ended up with 98% water in them, how much of their weight did they lose?

The cucumbers lost half of their weight.

If the water was 99% of the total weight, the remaining substance must have weighed 0.1lbs. If after the evaporation the substance comprises 2% = 1/50 of the cucumbers, the total weight must be 50 x 0.1lbs = 5lbs.

25 Horses

There are 25 horses and you want to find the fastest 3 among them. You can race any 5 of the horses against each other and see the final standing, but not the running times. If all the horses have constant, permanent speeds, how many races do you need to organize in order to find the fastest 3?

Let us label the horses H1, H2, H3, H4, …, H24, H25.

We race H1 – H5 and (without loss of generality) find that H1 > H2 > H3 > H4 > H5. We conclude that H4, H5 are not among the fastest 3.

We race H6 – H10 and (without loss of generality) find that H6 > H7 > H8 > H9 > H10. We conclude that H9, H10 are not among the fastest 3.

We race H11 – H15 and (without loss of generality) find that H11 > H12 > H13 > H14 > H15. We conclude that H14, H15 are not among the fastest 3.

We race H16 – H20 and (without loss of generality) find that H16 > H17 > H18 > H19 > H20. We conclude that H19, H20 are not among the fastest 3.

We race H21 – H25 and (without loss of generality) find that H21 > H22 > H23 > H24 > H25. We conclude that H24, H25 are not among the fastest 3.

We race H1, H6, H11, H16, H21 and (without loss of generality) find that H1 > H6 > H11 > H16 > H21. We conclude that H16, H21 are not among the fastest 3.

Now we know that H1 is the fastest horse and only H2, H3, H6, H7, H11 could complete the fastest three. We race them against each other and find which are the fastest two among them. We complete the task with only 7 races in total.

Donuts and Candies

Huey has 3 donuts, Dewey has 5 donuts. Louie comes along and three of them split the donuts equally. In exchange, Louie offers 8 candies to Huey and Dewey. What is the fair way to split the candies?

Huey must take 1 chocolate, and Dewey must take 7. This is because each of them ate 8/3 donuts, and therefore Huey gave away 1/3 of a donut, whereas Dewey gave away 7/3 of a donut.

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