Coins on a Chessboard

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?

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.

Odd Rectangle

The sides of a rectangle have lengths which are odd numbers. The rectangle is split into smaller rectangles with sides which have integer lengths. Show that there is a small rectangle, such that all distances between its sides and the sides of the large rectangle have the same parity, i.e. they are all even or they are all odd.

Source: Shortlist IMO 2017

Split the large rectangle into small 1×1 squares and color it in black and white, chessboard-style, such that the four corner squares are black. Since the large rectangle has more black squares than white squares, one of the smaller rectangles also must have more black squares than white squares. Therefore, the four corners of that smaller rectangle are all black. Then, it is easy to see that all distances between its sides and the sides of the large rectangle have the same parity.

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.

Source: MIT

Pay attention to the words in green. They form a riddle which needs to be answered.

Pay attention to the broken lines along the bubble speeches. Use an appropriate code to decode them.

Pay attention to the ship, the brick wall, the ladder, and the bucket. Use an appropriate code to decode them.

Pay attention to Grover’s arms. Use an appropriate code to decode them.

Pay attention to the fonts used for typing the words in red. Use their first letters to form a word.

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 Pokemons. Use their first letters to form a word.

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.

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.