Logic Puzzles
You have three light switches outside a closed room and three bulbs inside. You may flip switches any way you like, then enter the room once. How do you tell which switch controls which bulb?
Turn switch A on for a few minutes, turn it off, turn switch B on, go in.
You may add 1 to one number and subtract 1 from another. Starting with numbers (1,1,1,1), can you ever make one of them equal to 4 while the others remain nonnegative?
No.
A person says, “I am a liar.” Can that be true?
Impossible statement
You flip a fair coin 5 times. What’s more likely: getting exactly two heads or getting at least two heads? (Pick one and explain why.)
At least two heads is more likely.
Find the smallest positive integer that leaves remainder 1 when divided by 2, 3, 4, 5, and 6, and is divisible by 7.
301
A 4×4 chessboard has one corner square removed. Can you tile the remainder with dominoes (1×2)?
No.
A says, “At least one of us is a liar.” B stays quiet. Who’s who?
A is telling the truth, B is the liar.
You and a friend each pick a number from 1–10 (uniform). What’s probability you pick the same number?
1/10.
Fill 3×3 grid with 1–9 so every row and column has an odd sum
Not possible
You have 10 coins on a table, some heads, some tails. One move: flip any 3 coins. Show how to force all coins to heads, or prove impossible.
It depends on initial parity of tails; not always possible.
P1: “Exactly one of us tells the truth.”
P2: “P1 is lying.”
P3: “P2 is honest.”
P1 is honest, P2 and P3 are liars OR P2 is honest, P1 and P3 are liars.
There are 6 people in a room. Prove that two people must have the same number of friends in that room (friendship is mutual).
Pigeonhole principle → yes.
You and a friend take turns removing 1–3 stones from a pile of 21. The player who takes the last stone wins. You go first. What strategy guarantees a win?
First take 1, then always make the pair-sum to 4 your move (i.e., if opponent takes x, you take 4−x)
100 people stand in a circle and every minute each person simultaneously swaps hats with the person on their immediate right. After how many minutes does everyone have their original hat back?
100 minutes.
A: “B is lying.”
B: “C is lying.”
C: “A is honest.”
Only one sentence is true. Who’s who?
A: Liar
B: Honest
C: Liar
You shuffle a deck and draw 4 cards. What’s probability all 4 are the same suit?
1.05%
There are 5 houses in a row, each painted a different color. Five different pets live there (dog, cat, bird, fish, rabbit). Given these clues, who owns the fish?
The fish is owned in house 3
Start with the number 1 on the board. Allowed operation: replace n with n+1 or replace n with 2n. Can you reach 2019? If yes, find minimal number of moves.
Yes; 17 moves total.
Ten people wear red or blue hats. They can see everyone else’s hats but not their own. They all guess their own color at the same time. Can they plan a way so at least 5 are always right?
Yes - using parity (odd/even logic)
You have 5 identical-looking boxes; one box contains 2 red and 8 blue marbles, others contain 1 red and 9 blue. You pick a box uniformly at random and draw one marble (without looking). It’s red. What’s probability you picked the special box?
1/3