Conventional Riddles

Magic Square

3 x 3 square, all triples (across, down, and diagonal) must sum to 15. Use 1-9 once each.

Mutilated Chessboard Problem

Two tiles, each from opposite corners of a chess board, are removed from the board. Is it possible for 31 dominoes, each covering 2 full tiles, to completely cover the remaining tiles on the board? [Solve without the most obvious method]

Mutilated Cube Problem

Suppose you have 27 cubes stuck together, forming one larger (3 x 3 x 3) cube. What is the minimum number of straight cuts needed in order to separate all cubes from each other? After each cut, the pieces may be rearranged so that the next cut might separate more cubes. Prove your answer.

Some Knights

How many knights can you place on a chessboard such that none of them can attack another?

Two Ropes

You have two ropes, each of which will burn at varying speeds throughout their lengths, but by the end have burned for exactly 1 hour. How can you measure 45 minutes?

Hard Boiled

You have a 7-minute hourglass and an 11-minute hourglass, how can you boil an egg in exactly 15 minutes?

The Great Divide

Four people arrive at a river with a narrow bridge that can only hold two people at a time. It’s nighttime and they have one torch that has to be used when crossing the bridge. Person A can cross the bridge in one minute, B in two minutes, C in five minutes, and D in eight minutes. When two people cross the bridge together, they must move at the slower person’s pace. Can they all get across the bridge in 15 minutes or less?

A Very Contrived Situation

You buy a mansion. Right when you go inside, a robber jumps in through a window. He sees you and asks you to tell him which of the 3 switches in front of him turns on the attic light. If you lie to him, he will kill you. He lets you walk up the stairs once, but only once. Because you just bought the house, how will you know which switch turns on the attic light bulb?

Cheryl’s Birthday

Albert and Bernard just become friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates:

Cheryl then tells Albert and Bernard separately the month and the day of her birthday respectively.

Albert: I don’t know when Cheryl’s birthday is, but I know that Bernard doesn’t know too.
Bernard: At first I don’t know when Cheryl’s birthday is, but I know now.
Albert: Then I also know when Cheryl’s birthday is.

So when is Cheryl’s birthday?

Reservior Golfers

Four golfers named Mr. Black, Mr. White, Mr. Brown and Mr. Blue were competing in a tournament. The caddy didn’t know their names, so he asked them. One of them, Mr. Brown, told a lie.

The 1st golfer said “The 2nd Golfer is Mr. Black.”
The 2nd golfer said “I am not Mr. Blue!”
The 3rd golfer said “Mr. White? That’s the 4th golfer.”
The 4th golfer remained silent.

Which is which?

Sweet Boxes

There are three opaque boxes. One with mints in it, another with aniseed sweets and in the third a mixture of the two. They are all labeled incorrectly. What is the minimum number of sweets you can take to work out which box is which?

The Gold Scale

There are 100 bags each with 100 coins, but only one of these bags has gold coins in it. The gold coin has weight of 1.01 grams and the other coins has weight of 1 gram. We are given a digital scale, but we can only use it once. How can we identify the bag of gold coins?

Billiard Balls

This one’s a wee bit ridiculous You have 12 billiard balls, one is heavier or lighter than the rest. You can use a justice-y scale up to 3 times to determine which ball is the odd one, and whether it’s heavier or lighter.

The Emperor’s New Game

You are a prisoner sentenced to death. The Emperor offers you a chance to live by playing a simple game. He gives you 50 black marbles, 50 white marbles and 2 empty bowls. He then says, ‘Divide these 100 marbles into these 2 bowls. You can divide them any way you like as long as you use all the marbles. Then I will blindfold you and mix the bowls around. You then can choose one bowl and remove ONE marble. If the marble is WHITE you will live, but if the marble is BLACK… you will die.’ How do you divide the marbles up so that you have the greatest probability of choosing a WHITE marble?

Island Fire

On island trapped by mountains, totally forest, starts on fire in west, wind blows it east until it burn everything on island, how survive?

Two Guards

There are two paths forward, one will lead you where you want to go, the other will lead you to certain death. In front of each path stands a guard, one always lies, and the other always tells the truth. You don’t know which guard is which, nor which path is which. In one question, what can you ask one of the guards to find out which path will where you where you wish to go?

Spot the Dog

Spot is a dog. Every dog has one alpha (also a dog), and no two dogs have the same alpha. But Spot isn’t anybody’s alpha. :( For any two dogs, there is an assigned referee in case they get into a fight and an assigned marriage counselor in case they get married. The dogs have set up the following rules for deciding who will be the referees and counselors for who: If Spot fights with any dog, the other dog gets to be the referee. The referee of a fight between (dog 1) and (dog 2’s alpha) has to be the alpha of the referee of a fight between dogs 1 and 2. Spot has to be his own marriage counselor, no matter who he marries. The marriage counselor for (dog 1) and (dog 2’s alpha) has to referee any fight between dog 1 and the marriage counselor for dog 1 and dog 2. Who’s the marriage counselor for Spot’s alpha and the referee of a fight between Spot’s alpha and himself (Spot’s alpha)?

The Sum and Product Puzzle

$x$ and $y$ are two different whole numbers greater than $1$. Their sum is not greater than $100$, and $y$ is greater than $x$. Person $S$ knows the sum $x + y$, and person $P$ knows the product $x \times y$. Both $S$ and $P$ know all the information in this paragraph. The following conversation occurs:

$S$: “$P$ does not know $x$ and $y$”
$P$: “Now I know $x$ and $y$”
$S$: “Now I know $x$ and $y$”

What are $x$ and $y$?

import math

X_MIN   = 2
X_MAX   = 49
Y_MIN   = 3
Y_MAX   = 98
SUM_MAX = 100

# Return all valid pairs of numbers that sum to n
def sum_pairs(n):
    pairs = []
    for a in range(X_MIN, X_MAX + 1):
        b = n - a
        if b <= a:
            break
        pairs.append((a, b))
    return pairs

# Return all valid pairs of factors of n
def factor_pairs(n):
    lower_bound = X_MIN
    upper_bound = math.floor(math.sqrt(n))
    pairs = []
    for p in range(lower_bound, upper_bound + 1):
        if n % p == 0:
            q = n // p
            if q > Y_MAX:
                continue
            if p + q > SUM_MAX:
                continue
            if p == q:
                continue
            pairs.append((p, q))
    return pairs

# S: “P does not know X and Y”
# This implies that among all of the possible pairs of numbers summing to S's number,
# none of those pairs multiply to a number with just one valid pair of factors.
possible_pairs = set()
for sum in range(SUM_MAX + 1):
    pairs = sum_pairs(sum)
    valid = True
    for (x, y) in pairs:
        if (len(factor_pairs(x * y)) == 1):
            valid = False
            break
    if valid:
        possible_pairs.update(pairs)

# P: “Now I know X and Y”
# This implies that P's number did have multiple valid factor pairs, 
# but only one pair appeared in the list resulting from S's first statement.
possible_products = set()
duplicate_products = set()
for (x, y) in possible_pairs:
    if x * y in possible_products:
        duplicate_products.add(x * y)
    else:
        possible_products.add(x * y)

unique_product_pairs = set()
for (x, y) in possible_pairs:
    if x * y not in duplicate_products:
        unique_product_pairs.add((x, y))

# S: “Now I know X and Y”
# This implies that S's number did have multiple valid sum pairs, 
# but all of those pairs except for one were ruled out by their presence
# in the set of duplicate product pairs we obtained from P's statement.
possible_sums = set()
duplicate_sums = set()
for (x, y) in unique_product_pairs:
    if x + y in possible_sums:
        duplicate_sums.add(x + y)
    else:
        possible_sums.add(x + y)

unique_sum_pairs = set()
for (x, y) in unique_product_pairs:
    if x + y not in duplicate_sums:
        unique_sum_pairs.add((x, y))

# The fact that we are able to find X and Y implies that the set of pairs
# has been reduced to one. This is the solution.
print(unique_sum_pairs)

The Impossible Chessboard Problem

The jailer will take you into a private cell. In the cell will be a chessboard and a jar containing 64 coins.

The jailer will take the coins, one-by-one, and place a coin on each square on the board. He will place the coins randomly on the board. Some coins will be heads, and some tails (or maybe they will be all heads, or all tails; you have no idea. It’s all at the jailers whim. He may elect to look and choose to make a pattern himself, he may toss them placing them the way they land, he might look at them as he places them, he might not …). All you can do is watch.

Once all the coins have been laid out, the jailer will point to one of the squares on the board, indicating the one with the key under it (invisible from your position). The jailer will then allow you to turn over any one coin on the board. This is the only change you are allowed to make to the jailers initial layout. You will then be lead out of the room, and the jailer will bring in your friend, who will examine the board of coins and decide under which square the key is. They have only one chance to guess correctly, in which case you’re both set free.

The jailer explains all these rules, to both you and your friend, beforehand and then gives you time to confer with each other to devise a strategy for which coin to flip.