Division Impossible
Can you divide the following shape into 4 identical pieces?
Remark: The pattern is not important.
SOLUTION
Divide the piece into 4 smaller pieces that have the same shape and half dimensions.
Category: Insight
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Heaven or Hell
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.
Crashing Light Bulbs
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.
9 balls, 1 defective
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.
Drown or Burn
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.
The Next Die
Which die completes the following sequence:
SOLUTION
If you look at dots in the top row, you will see that they are put together into groups of 1, 2, 3, 4, and so on.
Buffalo Buffalo Buffalo
The sentence below is grammatically correct. Can you explain it?
Buffalo buffalo Buffalo buffalo buffalo buffalo Buffalo buffalo.
SOLUTION
The sentence says that buffalo (animals) from Buffalo (city, US), which are buffaloed (intimidated) by Buffalo (city, US) buffalo (animals), themselves buffalo (intimidate) buffalo (animals) from Buffalo (city, US).
Rhombuses
A regular hexagon is split into small equilateral triangles and then the triangles are paired arbitrarily into rhombuses. Show that this results into three types of rhombuses based on orientation, with equal number of rhombuses from each type.
SOLUTION
Color the rhombuses based on their type and imagine the diagram represents a structure of small cubes arranged in a larger cube. If you look at the large cube from three different angles, you will see exactly the three types of rhombuses on the diagram.
Alternatively, the problem can be proven more rigorously by considering the three sets of non-intersecting broken lines connecting the pairs of opposite sides of the hexagon, as shown on the image below. The type of each rhombus is determined by the types of the broken lines passing through it. Therefore, there are n² rhombuses of each type, where n is the length of the hexagon’s sides.
Unravel the Rope
Is it true that for every closed curve in the plane, you can use a rope to recreate the layout, so that the rope can be untangled?
Said otherwise, you have to determine at each intersection point of the closed curve, which of the two parts goes over and which one goes under, so that there aren’t any knots in the resulting rope.
SOLUTION
Start from any point of the curve and keep moving along it, so that at each non-visited intersection you go over, until you get back to where you started from.
Puzzle at the End of the Book
“Puzzle at the End of the Book” is a very challenging puzzle from the 2017 MIT mystery hunt. The answer to this puzzle is a 6-letter word, related to a woman’s beauty. The solution is intricate and requires careful analysis of the book, some geeky references, and possibly a good amount of Google searching. Use the hints below if you need help with solving puzzle.
Page 1 
Page 2 
Page 3 
Page 4 
Page 5 
Page 6 
Page 7 
Page 8 
Page 9
Source: MIT
HINT 1
Pay attention to the words in green. They form a riddle which needs to be answered.
HINT 2
Pay attention to the broken lines along the bubble speeches. Use an appropriate code to decode them.
HINT 3
Pay attention to the ship, the brick wall, the ladder, and the bucket. Use an appropriate code to decode them.
HINT 4
Pay attention to Grover’s arms. Use an appropriate code to decode them.
HINT 5
Pay attention to the fonts used for typing the words in red. Use their first letters to form a word.
HINT 6
Pay attention to the unusual words appearing in the text. Use parts of these words, combined with immediately preceding/succeeding parts of neighboring words, to get the names of six
HINT 7
The names of the six muppets have the same lengths as the six words discovered from the previous steps. See which letters overlap when you compare each muppet name with its corresponding word. Arrange these letters to get the final answer.
SOLUTION
The answer to this puzzle is MAKEUP.
In order to get to it, first you must find 6 secret fantasy related words.
1. The green words on the pages of the book form the sentence Wooden ship turned around before understanding sea monster (SIX). “Wooden ship” = ARK, “turned around” -> KRA, “understanding” = KEN, so we get KRAKEN, which is a sea monster with six letters.
2. The broken lines along the speech bubbles can be decoded using Morse code to spell Lilith, Morrigan, Scarlet, or Queen of Pain. These female demons give the secret word SUCCUBUS.
3. The ship, the brick wall, the ladder, and the bucket contain four hidden Brail letters, which spell out the word HUMA.
4. Grover’s arms encode through semaphore the Inuit mythological creature QALUPALIK.
5. The word “Puzzle” is written in five different fonts – Times New Roman, Impact, Twentieth Century, Arial, Nosifer. The first letters of these fonts form the word TITAN.
6. Each page from 2 to 8 contains some unusual words. Part of these words, combined with immediately preceding/succeeding parts of neighboring ones, give the six Pokemons Sandshrew, Pinsir, Ekans, Clefairy, Tentacruel, Eevee, Rapidash. Their first letters form the secret word SPECTER.
The names of the six muppets on the last page are Barkley, Donmusic, Elmo, Kermit, Misspigy, Oscar. They perfectly match in terms of length with the six secret words which we found above. Also, each pair of name with secret word overlap in just one position, the six resulting letters are E, U, M, K, P, A. If we arrange these letters with respect to the length of their corresponding words, we get the final answer MAKEUP.