Indexable collection of logic riddles with full statement and explained solution.
You have 1000 bottles of wine. One of them is poisoned. One drop from the poisoned bottle kills a rat in exactly 24 hours. You have 10 lab rats. How much time do you need at MINIMUM to positively identify the poisoned bottle?
There are 100 prisoners numbered 1 to 100. In a room there are 100 closed boxes, also numbered 1 to 100.
There are 100 prisoners in line. They randomly put a blue or red hat on each one.
There are 100 prisoners. Each day, the guard chooses one at random and takes it to a room with a light bulb (with on/off switch).
You have 25 horses and a race track with 5 lanes. You need to find the 3 fastest horses.
You are in a contest with three closed doors. Behind one there is a car (prize), behind the other two there are goats.
Albert and Bernardo have just met Cheryl. Cheryl tells them: 'My birthday is one of these 10 days: May 15, May 16, May 19, June 17, June 18, July 14, July 16, August 14, August 15, August 17.' Then he tells Albert only the month, and Bernardo only the day.
Five perfectly rational and selfish pirates (A, B, C, D, E, in order of rank) find 100 gold coins. They must decide how to distribute them according to this rule: the highest ranking pirate proposes a distribution.
You are in a room with two doors. One leads to freedom, the other to death.
In a room there are three switches. In another room (which you can't see from where the switches are) there is a light bulb. You can only enter the light bulb room ONCE. How can you determine which switch controls the light bulb?
You have three closed boxes with labels: "Apples", "Oranges" and "Mixed". You know all three labels are wrong. You can take only one fruit from a single box (without looking inside first). How do you reliably correct all three tags?
A ship is anchored in the port. On its side there is a staircase with 10 steps visible over the water. Each step is 20 cm high. The tide rises at a rate of 15 cm per hour. How many hours will it take to cover 4 steps?
You have two jugs: one with 3 liters and one with 5 liters. You have a tap with unlimited water. How can you measure exactly 4 liters of water? There are no measurement marks on the jugs, you can just fill them completely or empty them completely.
Four people need to cross a bridge at night. They only have one flashlight and the bridge can only hold two people at a time.
You have two hourglasses: one for 7 minutes and one for 11 minutes. How can you measure exactly 15 minutes?
You have two watches: - Clock A: is completely stopped. - Clock B: works, but goes back exactly 1 minute every hour. Which of the two keeps the correct time more times?
Now I am 40 years old. I tell you: “I am 4 times the age you were when I was the age you are now.” How old are you currently?
Three perfectly logical wise men are sitting in a circle. The king shows them 5 hats: 3 white and 2 black.
Three gods A, B and C are called Truth, Falsehood and Random, in some order. Truth always tells the truth, False always lies, and Random answers randomly true or false.
Two mathematicians ride a bus and read the line number. One says to the other: 1. “I have at least two children.” 2. “Their ages are positive integers.” 3. “The sum of their ages is the number of this bus line.” 4. “The product of their ages is my age.” The other answers: > “If you told me your age and how many children you have, could I deduce their ages?” The first one answers: > “No.” After hearing that, the second says: > “Now I know your age.” What…
Two different integers $x,y$ satisfy $2 \le x < y \le 99$. You say to one person $S=x+y$ (they only know the sum) and to another $P=xy$ (they only know the product).
There are a million light bulbs in a row, all initially turned off. There are a million people.
Three perfectly logical prisoners are in separate cells. The guard tells them, "I'll give each of you a red or blue hat.
There are two prisoners and a guard. The guard has a $8\times 8$ chess board and a coin on each square (64 in total), each one heads or tails.
You have two ropes and a lighter. Each rope takes exactly 1 hour to burn completely, but they do NOT burn evenly (some parts burn faster than others). How can you measure exactly 45 minutes using these two strings?
You and I are far away and can only send trunks to each other by courier. - Each one has their own padlock and their own key. - If a trunk travels without a lock, what it carries can be stolen. - We do not have a shared secret key. How can I send you information so that the trunk never travels open and only you can read its contents at the end?
Draw 9 points forming a square of $3 \times 3$. Can you connect the 9 dots using only 4 continuous straight lines without lifting the pencil from the paper?
Three missionaries and three cannibals must cross a river with a boat that only carries a maximum of two people. On no shore can there be a group where the cannibals are more than the missionaries (if there are missionaries present), because then they devour them.
On an island there are three types of inhabitants: knights (they always tell the truth), squires (they always lie) and spies (they can lie or tell the truth). You meet three people: A, B and C.
A tennis tournament involves 1,024 players in a single-elimination format (the winner advances, the loser is eliminated). How many games do you need to play to determine the champion? Don't use formulas, think about the logic of the problem.
You have 12 coins identical to the naked eye. One is false and may be heavier or lighter (you don't know which of the two cases).
Why are manhole covers circular instead of square or any other shape? Explain technical reasons. Google Interview Classic
How many times a day do the minute and hour hands of an analog clock overlap exactly? (Don't count the overlap at 12:00 twice)
You have two identical eggs and access to a 100-story building. Eggs can be very resistant or very fragile. Both are identical. You need to determine the highest floor from which an egg can be dropped without breaking. What is the MINIMUM number of throws you need in the WORST case to guarantee you find the critical floor? Microsoft Classic
A rope surrounds the Earth adjusted to the equator. It then elongates exactly $2\pi$ meters and separates itself evenly from the ground around the entire planet. What is the new height of the rope above the surface? Does it depend on the size of the Earth?
A farmer must cross a river with a wolf, a goat and a cabbage. Your boat can only carry the farmer and one of the three elements. If you leave the wolf alone with the goat, the wolf will eat it. If you leave the goat alone with the cabbage, the goat will eat it. How can he cross the river with everything intact? Riddle documented in the Tang dynasty, 9th century
There are three rods and a tower of $n$ discs of different sizes, stacked from largest to smallest. You can only move one disk at a time and never put a large disk on top of a small disk. What is the minimum number of movements necessary to move the entire tower to another rod? Classic problem attributed to the Brahma puzzle
100 perfectly logical people live on an island. There are people with blue eyes and people with brown eyes, but no one knows what color their own eyes are (there are no mirrors).
A wise man has three clay vessels: one of 12 liters, one of 8 liters and one of 5 liters. The 12 liter vessel is filled with sacred oil. You must divide the oil into two exactly equal parts of 6 liters using only these three vessels. How do you do it? Problem of the Bakhshali Manuscript, 3rd-4th century
A father leaves 17 camels to his three sons with these conditions: the oldest gets $\frac{1}{2}$, the second gets $\frac{1}{3}$ and the youngest gets $\frac{1}{9}$. Splitting camels is not allowed. How can the distribution be made by exactly complying with the will? Classic recreational arithmetic puzzle in the Middle East
A farmer buys 100 animals for exactly 100 silver coins. Buffaloes cost 10 coins each, pigs cost 3 coins each, and chickens cost 0.5 coins each. Buy at least one of each type. How many animals of each type did you buy? Classical "Master Sun" problem, 3rd-5th century, China
A sultan has 9 pearls that look identical, but one is slightly heavier. It has a two-pan scale. You must identify the heaviest pearl by using the scale the MINIMUM number of times possible, because each use of the scale costs one gold coin. How many weighings do you need at least? 12th century Persian riddle
Place the numbers 1 through 9 on a $3 \times 3$ grid so that the sum of each row, each column, and each diagonal is exactly 15. This is the legendary magic square of Lo Shu, discovered according to legend in the shell of a divine turtle. What is the solution? How many unique solutions are there (not counting rotations and reflections)? Chinese mathematics, ~2200 BC
There are 10 bags with coins. In 9 bags each coin weighs 10 g. In 1 bag, they all weigh 11 g. With a single weighing on a digital scale, how do you identify the heavy bag?
A forklift only starts if the total load is exactly 100 kg or exactly 150 kg. You have 5 indivisible boxes with unknown weights. An operator wrote down the sums of all the pairs of boxes (the 10 possible pairs): $$ 110,\ 112,\ 113,\ 114,\ 115,\ 116,\ 117,\ 118,\ 120,\ 121. $$ Is there a selection of boxes (1, 2, 3, 4 or all 5) that allows the forklift to be activated?
There are three piles with 3, 4 and 5 chips. On each turn you can remove any positive number of tokens, but from a single pile. Misère rule: Whoever takes the last piece loses. Does the first player have a winning move? If yes, what should be your first move?
Four special dice: - A: 4, 4, 4, 4, 0, 0 - B: 3, 3, 3, 3, 3, 3 - C: 6, 6, 2, 2, 2, 2 - D: 5, 5, 5, 1, 1, 1 You choose a die first. The dealer then chooses another die. Whoever rolls the highest number wins. The dealer states: “no matter which one you choose, I can choose a die that favors me with probability $2/3$”. How can it be true? What is your strategy?
In a round robin tournament with $n$ players: - each pair plays exactly once, - there are no ties. Can you always order the players in a row $P_1,P_2,\dots,P_n$ such that each player has beaten the one to his right? That is to say: $$ P_1 \to P_2 \to \cdots \to P_n. $$
Five teams play a round-robin league, a single round. Rules: - victory: 3 points, - tie: 1 point per team, - defeat: 0 points. In the end, the five teams end up tied on points, and that common score is odd. How many games ended in a draw?
Crossing the entire desert requires exactly 6 days. Each explorer can carry supplies for a maximum of 4 days. Explorers can: - walk together, - transfer food, - and some may return to base. Condition: no one can run out of supplies at any time. What is the minimum number of explorers that must depart to ensure that exactly one completes the crossing?
There is a safe with a 4-digit code (0000 to 9999). You can type a long sequence of digits. The box opens as soon as the last 4 digits entered match the code. How many digits do you need to type at least to guarantee opening, without knowing the code?
An airplane has $n$ seats ($n\ge 2$) and $n$ passengers with assigned seats. Passenger 1 loses his card and sits in a random seat. From passenger 2 onwards: - if your seat is free, you sit in it; - if it's busy, choose a free seat at random. What is the probability that the last passenger sits in his own seat?
Four spies A, B, C and D each have a different secret. In each phone call, the two participants tell each other everything they know up to that moment.
You have 50 white balls and 50 black balls. You must distribute them in two boxes, complying with: - each box has at least one ball. Then this experiment is done: 1. one of the two boxes is chosen at random (probability 1/2 each), 2. A ball is drawn at random from that box. You win if the ball drawn is white. How should you distribute the balls to maximize the probability of winning? What is that maximum probability?
You want to color the numbers $\{1,2,3,4,5\}$ with two colors (red and blue). The prohibited rule is: - there cannot be three numbers of the same color $x,y,z$ that satisfy $x+y=z$ (allowing $x=y$). Question: is there any color that meets the rule?
Three sources activate every 4, 6 and 9 minutes. At 12:00 all three were activated at the same time. A visitor enters between 12:00 and 12:30. Since entering: - the first match from exactly two sources occurs at 2 minutes, - the second match from exactly two sources occurs at 8 minutes, - the first match of the three occurs at 20 minutes. What time did you come in?
Five books of different heights 1,2,3,4,5 (1 = shortest, 5 = tallest) are in a row. Each book writes down how many books taller than it are to its left. The notes, from left to right, are: $$ 0,\ 1,\ 1,\ 3,\ 0. $$ What is the order of heights from left to right?
Four bells A, B, C, D begin silently. There are four strings and each one changes (turns on/off) these bells: - X changes A and B, - And change B and C, - Z changes A, C and D, - W changes C and D. Each rope can be pulled at most once. In the end, only A and D are playing. What ropes were pulled?
Four distributors—Ana, Beto, Cora and Diego—came in positions from 1 to 4, without ties. Exactly one of these notes is known to be false: - Ana: “I was first.” - Beto: “Ana arrived immediately before me.” - Cora: “Ana arrived before Diego.” - Diego: “Beto arrived before Cora.” Determine the order of arrival.
You start with: $$ 1,\ 2,\ 3,\ 4,\ 5. $$ The only operation allowed is to choose three consecutive cards and rotate them cyclically: $$ abc\to bca\quad\text{o}\quad abc\to cab. $$ Can you reach: $$ 2,\ 1,\ 3,\ 4,\ 5? $$ > Didactic closure of the level > > The difference between “normal” and “misère” seems small, but it changes the closure of the strategy.
There are two coins in a bag: - Coin A: has a face on both sides. - Coin B: it is normal (heads and tails). You pull out a coin at random, flip it and it comes up heads. What is the probability that the other side of that same coin is also heads?
You start with 10 white chips on the table. In each movement you are obliged to: - choose exactly two tiles, - and flip them (white to black, black to white). Question: can you end up with exactly one black tile? > Clarification of the statement: you can repeat movements as many times as you want, but always turning over 2 pieces per turn.
An experimental traffic light has three lights: red ($R$), yellow ($A$), and green ($V$). Every minute, the state of the next minute is calculated like this: - $R$ is turned on if and only if $A$ was turned off. - $A$ is turned on if and only if $V$ was turned on. - $V$ is turned on if and only if $R$ was turned on. Initial state (minute 0): $$ (R,A,V)=(0,0,0), $$ where 0 = off and 1 = on. At what minute are all three turned off again for the first time…
On an 8x8 chess board you remove two opposite corners. Can the rest of the board be exactly covered with 1x2 dominoes, without overlapping or leaving gaps?
On an island there are: - 4 red chameleons, - 7 green, - 10 blue. When two of different colors are found, both change to the third color. 1. Is it possible to reach a state where everyone has the same color? 2. If possible, give a specific sequence and the minimum number of encounters.
There are 100 people. On each front an integer from 1 to 100 is written (they can be repeated).
There is an infinite line of whole squares: $$ \dots,-2,-1,0,1,2,\dots $$ An invisible submarine has: - unknown integer initial position $X$, - unknown constant integer velocity $V$ (can be negative, positive or zero). On turn $t=1,2,3,\dots$, you can shoot a single entire square. The submarine on turn $t$ is in: $$ X+Vt. $$ Is there a strategy that guarantees impact in a finite number of turns, regardless of $X$ and $V$?
On a $100\times100$ board, exactly 99 squares start out infected. Minute rule: - a healthy square is infected if it shares an orthogonal border with at least two infected squares, - an infected square is never cured. Question: Is it possible that, after enough time, all 10,000 cells become infected?
Initial urn: 1 red and 1 blue ball. In each turn: 1. a ball is drawn at random, 2. is returned to the urn, 3. and an extra ball of the same color is added. After $n$ turns there are $n+2$ balls. Two bets: - Carlos: “the proportion of red is concentrated around 1/2”. - Maria: “all possible final numbers of reds are equiprobable.” Who is right?
You have a scale with two pans and four weights: $$ 1,\ 3,\ 9,\ 27\ \text{kg}. $$ You can put weights on either side. How many different whole weights between 1 and 40 kg can you measure exactly?
There are 10 people in a row, numbered from 1 (front) to 10 (behind). Each hat can be red, blue or green. Vision and turn rules: - person 10 speaks first and sees the hats of 1..9, - person 9 speaks next and sees 1..8, -... - person 1 speaks last and sees none. Everyone hears the previous answers. Before starting you can agree on a strategy. How many people can be guaranteed to be saved with certainty?
There is a rectangular chocolate bar of $m\times n$ squares, with $m,n\ge2$. The box in the lower left corner is poisoned.
A spectator chooses 5 different cards from a standard deck of 52. The assistant sees the 5 cards, hands 4 to the magician (in the order he wants) and hides the fifth.
A code has four different digits. It is known that: - is a multiple of 9, - the last figure is the remainder by dividing the sum of the first three by 10, - By reversing its figures, the new number is 369 greater than the original. What is the code? > Didactic closure of the level > > A good expert solution combines two things: proof of possibility and explicit construction.
A courier must travel each street exactly once on the following map:
``text
A---B
|\ |
| \ |
C---D---F
\ | /
\ | /
E
``
To avoid ambiguity, the streets (edges) are exactly these 9:
- A-B, A-C, B-C, B-D, C-D, C-E, D-E, D-F, E-F.
Is there a route that uses all streets exactly once?
If it exists, it indicates which neighborhood it should start from, which one to end in, and gives a valid route.
You have a scale with two plates and three visually identical balls. You know that one ball weighs differently than the other two, but you don't know if it is heavier or lighter.
Two trains are 300 km apart and are approaching each other, each at 60 km/h. A fly leaves the first train towards the second at 90 km/h.
You climb a 12-step ladder. In each movement you can go up 1 or 2 steps. How many different ways can you get to the top?
There are 7 people sitting in a circle. Each one says exactly: > “My neighbor on the right is a liar.” It is known that each person is of a unique type: - or always tells the truth, - or always lies. Is it possible for all 7 phrases to be compatible at the same time?
At a party there are $n$ people, with $n\ge 2$. Each person writes down how many handshakes they gave during the party.
In a meeting of 6 people, exactly one of two things happens for each couple: - or they know each other, - or they don't know each other. Prove that at least one of these two situations always happens: 1. there are 3 people who mutually know each other; 2. There are 3 people who are mutually unknown to each other.
Search for a 10 digit number $$ d_0d_1d_2d_3d_4d_5d_6d_7d_8d_9 $$ such that: - $d_0$ = number of zeros in the number, - $d_1$ = number of ones, - $d_2$ = number of twos, -... - $d_9$ = number of nines. What is that number?
You have the row: $$ 4,\ 2,\ 6,\ 1,\ 5,\ 3. $$ The only operation allowed is to exchange two adjacent cards whose sum is odd. Can you reach: $$ 1,\ 2,\ 3,\ 4,\ 5,\ 6? $$
A 6 letter password was hashed. All its consecutive fragments of length 3 were preserved, out of order: $$ \texttt{ABA},\ \texttt{BAC},\ \texttt{ACB},\ \texttt{CBA}. $$ Reconstruct the original password. > Didactic closure of the level > > If you can justify why recurrence is correct and not just use it, you are already at a solid level.
Dispones de una balanza de dos platos y de cuatro pesas de 1 g, 3 g, 9 g y 27 g. En cada pesada puedes colocar cada pesa en el plato izquierdo, en el derecho o dejarla fuera. ¿Qué pesos enteros exactos puedes equilibrar con esta balanza?
Hay 100 monedas sobre una mesa. Sabes que exactamente 20 están cara arriba y 80 cara abajo, pero estás completamente a oscuras.
En una urna hay bolas blancas y negras. Repetidamente extraes dos bolas: si son del mismo color, retiras ambas y metes una negra; si son de distinto color, retiras ambas y metes una blanca.
Se numeran del 1 al n las posiciones de un círculo. Se elimina primero al 2, luego al 4, luego al 6, y así sucesivamente, continuando de manera circular entre las posiciones aún no eliminadas, hasta dejar una sola. ¿Qué número sobrevive en función de n?
Entre la casa de un mensajero y su destino hay siete jornadas. Al final de cada jornada hay una casa donde puede dormir.
Observa la sucesión:
Cuatro prisioneros (A, B, C y D) están alineados. Hay un muro entre C y D.