Photograph via snooOG

This subreddit is for anyone to share math or logic related riddles, and try and solve others. Come check it out! This subreddit is designed for viewing on old.reddit.com.

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Welcome to Math Riddles! Post your math and logic puzzles, and try and solve others! While the subreddit aims for math related riddles, all logic puzzles and riddles are welcome as well.


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challenge ideas for teenagers

Hi everyone!

I run a math and science competition at a summer camp for kids who are quite interested and advanced in STEM! Most days they are solving olympiad style problems, but there is one day where we do a more silly fun competition. I created this little challenge for them last year and was wondering if you guys had similar ideas that emulate competing for limited resources I would be interested in hearing them since I can't exactly repeat this one!

Challenge Rules:

Math Challenge: Math-themed Auction

The math challenge will be an auction, where you will buy various items to create a math expression. The items for sale will be both math symbols (x, +, -)  and numbers (such as 7, 23, 45). The goal is to win these items to create a math expression where the output is as close to 100 as possible.

You will start with 65 dollars, and there will be 6 rounds where 7 items are auctioned off each round. You can see the items for each round in the handout given to your teams. Each round also has a mystery item that we will announce when the round starts.

Auction Rules

Items will be sold through a blind Dutch Auction. This means that you cannot see how much the other teams are bidding. At the end of each round, the team with the highest for each item will win that item, and they pay the price of the second highest bid.

The total sum of how much you bid must not exceed the amount of money that you have left. If there is a tie for highest bid, the team which correctly answers a tiebreaker question first gets the item. If you are the only bidder for an item, you pay zero!

Math Expression

Once you have bought the items, you will use them to create your math expression. You can use the remaining amount of money that you have left as a number in your expression.

1 Comment
01:32 UTC


Consecutive Primes Puzzle

Find 3 consecutive prime numbers that can each be written as a sum of 3 consecutive primes, where each of these 3 sets of primes share one element in common that can also be written as a sum of 3 consecutive primes.

09:00 UTC


just another easy expected value problem

randomly permute n distinct integers. what is the expected number of local maximum?

an integer is a local maximum iff it is greater than all its neighbors. eg: 2,1,4,3 has two local max: 2 and 4.

unrelated note: apparently this is an interview problem, from where a friend told me.

03:07 UTC


Impossible fish problem

Let's say there's a fish floating in infinite space.


You only get one swipe to catch it with a fishing net.

Which net gives you the best odds of catching the fish:

A) 4-foot diameter net

B) 5-foot diameter net

C) They're the same odds

Argument for B): Since it's possible to catch the fish, you obviously want to use the biggest net to maximize the odds of catching it.

Argument for C): Any percent chance divided by infinity is equal to 0. So both nets have the same odds.

Is this an impossible question to solve?

10:24 UTC


just another bit flipping game

in m x n board, every square is either 0 or 1. the goal state is to perform actions such that all square has equal value, either 0 or 1. the actions are: pick any square, bit flip that square along with all column and row containing that square.

we say m x n is solvable if no matter the initial state, the goal state is always reachable. so 2 x 2 is solvable, but 1 x n is not solvable for n > 1.

for which m,n ∈ Z+ such that m x n is solvable?

11:21 UTC


Sum of Digital Powers

Let T be the set of positive integers with n-digits equal to the sum of the n-th powers of their digits.

Examples: 153 = 1^3 + 5^3 + 3^3 and 8208 = 8^4 + 2^4 + 0^4 + 8^4.

Is the cardinality of T finite or infinite?

21:55 UTC


Triangular Split Perfect Numbers

Let T_n = n(n+1)/2, be the nth triangle number, where n is a postive integer.

A split perfect number is a positive integer whose divisors can be partitioned into two disjoint sets with equal sum.

Example: 48 is split perfect since: 1 + 3 + 4 + 6 + 8 + 16 + 24 = 2 + 12 + 48.

For which n is T_n a split perfect number?

21:27 UTC


Triangular Perfect Numbers

Let T_n = n(n+1)/2, be the nth triangle number, where n is a positive integer.

A perfect number is a positive integer equal to the sum of its proper divisors.

For which n is T_n an even perfect number?

21:10 UTC


Four Dogs in a Field

Four dogs are at the corners of a square field. Each dog simultaneously spots the dog in the corner to her right, and runs toward that dog, always pointing directly toward her. All the dogs run at the same speed and finally meet in the center of the field. How far did each dog run?

18:03 UTC


No Four in Plane

On a 2x2x2 grid you can choose 5 points such that no subset of 4 points lay on a common plane. What is the most number of points you can choose on a 3x3x3 grid such that no subset of 4 points lay on a common plane? What about a 4x4x4 grid?

01:56 UTC


Exponential Polynomials

Let b be a positive integer greater than 1.

Let P_n be the unique n-degree polynomial such that P_n(k) = b^k for k in {0,1,2,...,n}.

Find P_n(n+1).

22:02 UTC


The Clock Triangle

Let the face of an analog clock be a unit circle. Let each of the clocks three hands (hour, minute, and second) have unit length. Let H,M,S be the points where the hands of the clock meet the unit circle. Let T be the triangle formed by the points H,M,S. At what time does T have maximum area?

18:35 UTC


Sum of Cubes of Digits

Find all positive integers that are the sum of the cubes of their digits.

18:24 UTC


Factorial Polynomials

Let P_n be the unique n-degree polynomial such that P_n(k) = k! for k in {0,1,2,...,n}.

Find P_n(n+1).

18:07 UTC


This vlogger vlogs till they die, 366 times.

Setup: A vlogger wants to record a vlog on a set interval i.e every subsequent vlog will be the same number of days apart. However they also want one vlog post for every day of the year.

They first came up with the solution to vlog every day. But it was too much work. Instead the vlogger only wants to do 366 vlogs total, and they want to vlog for the rest of their life.

Assuming the vlogger starts vlogging on or after June 16th 2024 and will die on January 1st 2070, is there a specific interval between vlogs that will satisfy all of the conditions? FWIW The vlogger lives in Iceland and where UTC±00:00 (Greenwich mean time) is observed year round.

  • 366 total vlogs
  • solve for vlog interval
  • 16,635 total days for vlog to take place.
  • The first Vlog must start on or after June 16th 2024 (but no later than the chosen interval after June 16th 2024)
  • The first possible vlog day is June 16th 2024
  • No vlogs may take place on January 1st 2070 or after (because the vlogger dies)
  • leap years are 2028, 2032, 2036, 2040, 2044, 2048, 2052, 2056, 2060, 2064, 2068

Tell me the date of the first vlog, and the interval. If this isn't possible I'm also interested in why!

I'm not that good at math and thought this would be an fun problem. I figured a mod function could be useful. If you think you can solve this problem without leap years please include your solution. As well if you can solve this problem without worrying about lifespan but have an equations that finds numbers that solve for a interval hitting every day of the year please include as well.

EDIT: DATE RANGE CLARIFICATION 16,635 total days. from and including: June 16 2024 To, but not including January 1, 2070

EDIT 2: Less than whole day intervals are okay! You can do decimal or hours or minutes. Iceland was chosen for being a very simple time zone with no daylight savings.

21:41 UTC


Virus vs Bacteria

A colony of n bacteria is invaded by a single virus. During the first minute it kills one bacterium and then divides into two new viruses; at the same time each of the remaining bacteria also divides into two. During the next minute each of the two newly born viruses kills a bacterium and then both viruses and all the remaining bacteria divide again, and so on. How long will the colony live?

11:36 UTC


A logical puzzle I can't wrap my head around.

Tne first version of this puzzle is from the 1930s by British puzzler Henry Ernest Dudeney. This one is a bit different though.

Here it goes:

Smit, Jones, and Robinson work on a train as an engineer, conductor, and brakeman, respectively. Their professions are not necessarily listed in order corresponding to their surnames. There are three passengers on the train with the same surnames as the employees. Next to the passengers' surnames will be noted with "Mr." (mister).

The following facts are known about them:

Smit, Jones, and Robinson:

Mr. Robinson lives in Los Angeles.
The conductor lives in Omaha.
Mr. Jones has long forgotten all the algebra he learned in school.
A passenger, whose surname is the same as the conductor's, lives in Chicago.
The conductor and one of the passengers, a specialist in mathematical physics, attend the same church.
Smit always beats the brakeman at billiards.

What is the surname of the engineer?

20:39 UTC


Number of distinct cubes with face diagonals

Imagine a cube where a diagonal line has been drawn on each face. As there are 6 faces, there are 2^(6) = 64 possibilities to draw these lines. How many of these 64 possibilities are actually distinct, i.e. cannot be transformed/rotated into one another?

05:22 UTC


just another simple number theory

Construct graph G(n,m) with n nodes, labeled 0 to (n-1). Connect each node k with node (m·k mod n) with undirected edge.

State the criteria for n ∈ Z^(+) and m ∈ Z such that the graph G(n,m) is connected, proof your statement.

03:26 UTC


just another simple problem

construct a long sequence with n distinct integers, such that all adjacent product are also distinct.

eg: for n=2, the longest sequence is 6,6,7,7 (not unique) , which has length of 4.

what is the longest sequence for each n?

bonus: what about cycles? for n=1 and 2 the longest cycle length is 1.

03:26 UTC


Game with 3 coins

I was sitting in my desk when my daughter (13 year old) approach and stare at 3 coins I had next to me.

1 of $1 1 of $2 1 of $5

And she takes one ($1) and says "ONE"

Then she leaves the coin and grabs the coin ($2) and says "TWO"

The proceeds to grab the ($1) coin and says "THREE because 1 plus 2 equals 3"

She drop the coins and takes the $5 coin and the $1 coin and says "FOUR, because 5 minus 1 equals 4"

She grabs only the $5 and says "FIVE "

then SIX


Then... She asked me... How can you do TWELVE?

So the rules are simple:

Using ANY math operation (plus, minus, square root, etc etc etc.)

And without using more than once each coin.

How do you do a TWELVE?

19:43 UTC


Infinite 15 puzzle

Consider an infinite grid of squares, where all rows and columns can be independently shifted (illustration on 6x6 grid). A valid sequence of moves is a possibly infinite sequence of shifts in which each individual square moves only a finite number of times.

Does there exist a valid sequence of moves which swaps adjacent squares? What about one which reflects all squares over the horizontal axis?

14:13 UTC


Boba Riddle

                  - READY IN 5 MINUTES -

You want to make black sugar flavor tapioca pearl boba and the container bag says the serving size is 1/3 cups (50g). There are 5 full servings in the container.

~Add 10 cups of water for every 1 cup of tapioca pearls in a large pot. Add tapioca slowly into boiling pot and stir lightly.

~Wait until tapioca floats to water surface. Cover pot. Cook in medium heat for 2-3 minutes. Turn off heat and simmer another 2-3 minutes (adjust time to soften tapioca to personal tastes).

-Scoop out tapioca and let it rest in cold water for 20 seconds. Scoop out tapioca into dry bowl and mix in sugar or honey.

  1. How many grams are in the container?
  2. How many cups are in the container?
  3. If you want only 1 serving, what is a correct estimate of how many cups you should boil?

a. If you only want 2 servings b. If you only want 3 servings c. If you want 4 servings d. If you want all 5 servings

Show your formula for all answers or/ how you got your answer.

Then enjoy your imaginary boba while I eat mine!

04:59 UTC


Casino Puzzle 🎲🎯

Here is a puzzle for those of you that are interested:

You're at a casino, and you have a number of chips. Each chip gives you a 20% chance at hitting a jackpot. Each chip costs 1/5th of the jackpot. Every round you can place a certain number of chips. 1, 2, 3, 4 or 5. The objective is to attain the highest possible balance. Placing 5 chips yields the same result as not participating.

Is the game statistically profitable to participate in? If so, what would be the ideal playing strategy?

19:38 UTC


Tournament Chain

In a tournament with 2^n teams, all teams played against each other. Show that we can find a list of n+1 teams, t_0, t_1, …, t_n such that t_i won against t_j for every 0 <= i < j <= n

Source: Quantum problem M66

10:05 UTC


An Interesting Tree Puzzle

100 trees are planted in the ground. Each tree grows at a different rate and will continue to grow until it dies, which can happen at any time. Some trees will die shortly after being planted while others will grow hundreds of feet tall.

You have the ability to bet on each tree’s growth.

Your bet scales directly with the tree’s growth but when the tree dies, your bet goes to $0. 


A tree is currently 1 foot tall and you bet $10 on it. For every foot that the tree grows, you will make $10.

What would you do to make money consistently in a situation like this?


  1. The maximum height each tree reaches before it dies follows a power law distribution
  2. Each tree starts out at 1 ft tall and you can place your bet at any time
  3. The growth rate of each tree is not the same


16:50 UTC


Amoeba Population Bomb

Amoebas reproduce by splitting into two, and the time to splitting events follow a Poisson distribution. Let p be the probability that one amoeba splits at least once in time t. If the initial population has A amoebas, what is the probability of at least B >= A amoebas at time t?

Inspired by this other problem. Spoilers at the link.

Hint: >!it's equal to the probability that if we toss B-1 biased coins, each with probability p of coming up heads, that we will get at least B-A heads!<

20:14 UTC


An extension problem

Let n >= 2. Suppose f: R^n -> R is continuous, and further is k-times continuously differentiable on R^n \ {x} for some point x. Assume also that the limit as we approach x of the k-th order derivative exists.

Show that f is in fact k times continuously differentiable on R^(n).

12:25 UTC


A curious contraction

Show there exists a strict contraction f on [0, 1] (i.e. |f(x) - f(y)| < |x - y| for all x =/= y) with |f’| = 1 almost everywhere.

23:43 UTC


Duplicating balls

There are a few white balls and one black ball in an (infinitely big) urn. Every turn, a ball is drawn from the urn uniformly at random. If a white ball is drawn, it is put back into the urn along with one more white ball. If a black ball is drawn, it is put back into the urn along with two more black balls.

Show that that no matter how many white balls we start with, we have that the ratio of black to white balls tends to infinity almost surely.

10:10 UTC

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