You play three tennis matches. You have a 70% chance of beating your teacher in any match and a 30% chance of beating a champion. Assume the results of all three matches are independent, and the win probabilities stay fixed.

Your goal is to win at least two consecutive matches. You can choose one of these orders:

Teacher — Champion — Teacher

Champion — Teacher — Champion

Which order gives the higher probability of achieving your goal, and why? Winning all three also counts as success.

Bonus: if the two win probabilities are any 0 < q < p < 1, is the same choice always better?

Creator disclosure: I run Think in Odds, a daily probability puzzle site. This is a released puzzle version; optional hints and a worked explanation are here: https://thinkinodds.com/puzzles/tennis?utm_source=reddit-mathriddles&utm_medium=reddit&utm_campaign=r_mathriddles_first_players_oct2026

A Logician writes three numbers on 3 separate cards and gives them to his 3 students.

He says," The 3 numbers are single digit prime numbers. Any combination. None of you know the other 2 numbers. But you can ask me one question that must start with "Is the SUM of the three numbers–” which I can only answer Yes or No. Given that info you can then declare that you know the other 2 numbers and/or who has them. OK?" 

Raj was first. He looked at his number and asked," Is the sum of three numbers an odd number?"

The Logician " No" 

Then Ken looked at his number and asked," Is the sum of the three numbers divisible by 4?"

The Logician said "Yes"

Lisa looked at her number and said,"Well, I know the other 2 numbers but cannot tell who has what number".

Raj then cheerfully said," I know who has what !" Ken said,” So do I” They then laid out the answer.

What were the three numbers? What number did Lisa have?

the image shows 3 rectangular grids, each partitioned into two regions. each region is connected and does not contain 2x2 subregion. unfortunately the third grid is partially destroyed. how many blue tiles are there?

Set A is called brain-rot iff it satisfies two conditions:

1. sum(A) is divisible by 10.
2. if 1∈A, then both 6,7∈A.

How many subsets of {1,2,…,100} is brain-rot?

Source: my rotten brain

A teacher writes six words on the board: CAT, DOG, HAS, MAX, DIM, TAG.

Then he hands three pieces of paper to three of his students: one to Alex, another to Ben, and another to Chris. The teacher explains that he has secretly chosen one of the words on the board, and has written on each piece of paper a different letter from that word. Students may look only at their paper and must not tell each other what letter they have.

After that the teacher says: "Everybody, please have a look at your letter and raise your hand as soon as you think you know the chosen word."

Alex immediately raises his hand.

Ben, after thinking for a while, also raises his hand.

Chris does not raise his hand.

The teacher then asks Alex: "Do you know which letter Chris has?"

"No, I don't" - says Alex.

Hearing that, Chris finally raises his hand.

Alex, Ben and Chris always ace their logic exams. What is the secret word?

(Came up with this variation of an old riddle and wanted workshop it here. EDIT: added my proposed solutin in the comments)

DÍGITO CERO

Se me ocurrió este reto matemático y quiero ver qué tan difícil puede llegar a ser.

OBJETIVO

Elige cualquier número entero del 1 al 1000 y consigue llegar exactamente a 0 usando la menor cantidad posible de operaciones.

REGLAS

1. No se puede repetir ningún dígito en toda la solución. Si aparece un 7 una vez, no puede aparecer otro 7.
2. No se puede multiplicar por 0.
3. Un resultado intermedio no puede volver a utilizarse después.
4. Cada operación debe ser simple y no puede contener otra operación dentro. Válido: √9 No válido: √(9-4)
5. El resultado de una operación no puede contener un dígito que ya aparecía en esa misma operación. Válido: 8 - 3 = 5 El 5 no aparecía antes en esa operación.

La meta no es simplemente encontrar una solución.

La meta es encontrar la solución con MENOS OPERACIONES POSIBLES.

Por eso también me interesa saber:

• ¿Cuál es el número más difícil?
• ¿Cuántas operaciones mínimas necesita cada número?
• ¿Hay números entre 1 y 1000 que no tengan ninguna solución?

El reto se llama:

DÍGITO CERO

El objetivo es llegar al 0.
Los dígitos no se pueden desperdiciar.
Y cada operación debe aportar algo nuevo.

A ver quién encuentra la solución más corta.

Let n be a fixed positive integer. Alice and Bob play the following game on the integer number line. Alice starts at 0 and Bob starts at n. They take turns making moves. On the i^th turn,

1) If i is odd, Alice moves to an integer at most 2^i -1 distance away from her current position.

2) If i is even, Bob moves to an integer at most 2^i -1 distance away from his current position.

Note that both players have the option to stay where they are on their turn. The game ends only when one player moves to the same position as the other player, in which case the player who moved wins. Find all positive integers n for which Alice has a winning strategy, and find all positive integers n for which Bob has a winning strategy.

Given number 479 made using the matchstck referenced in the figure above. Move exactly 2 matchsticks and create the largest possible number. You cannot change the format of the numbers shown in the reference. (For example you can only construct the number 1 with TWO matchsticks or 6 or 9 with 6 matchsticks). You can use any math operation known in standard math. The final number could be the result of this operation also. (For example, you can use the 2 matchsticks to create a multiplication operator x.)

The obvious answer obtained by exponenciation function might be way lower than another creative answer given to me by my mathematician friend.

Seven cells hold letters, three hold numbers. A five-letter English word is hidden, and it uses exactly five of the seven letters. The remaining two belong to no word and are pure noise.

The rule for a number: it equals the count of cells adjacent to it whose letter belongs to the hidden word. A letter outside the word contributes nothing to any number, so adjacency to it is invisible.

Adjacency, stated precisely, because this is the part that is easy to get wrong. Put the middle column on even y and the two side columns on odd y. Two cells are adjacent when they share a column and differ by 2 in y, or when their columns differ by 1 and their y differ by 1. On this ten-cell comb that gives two cells of degree 6, two of degree 4, and six of degree 3.

      ___
  ___/ H ___
 / 2 ___/ T \
 ___/ U ___/
 / 1 ___/ M \
 ___/ 4 ___/
 / R ___/ A \
 ___/ N ___/
     ___/

Which five letters, and which word?

What makes it worth the minute: the seven letters spell two perfectly ordinary five-letter words, so no amount of vocabulary settles it, and exactly one of the three numbers separates them. All three numbers are load-bearing. Withhold any single one and the count of consistent five-letter subsets goes from 1 to 4.

Uniqueness is exhaustive over the 21 subsets, not argued by eye.

Spoiler tag your answer.

As I was falling asleep last night, I thought it was kinda cool that 56 = 7 * 8 works in base 10, specifically how it consists of four consecutive digits in order. Then I realized it actually happens again! 12 = 3 * 4

Is there any other base such that there are four consecutive digits A, B, C, D (in increasing order) such that AB = C * D? If so, are there any (besides base 10) where it happens twice? Why or why not?

Thank you for the warm comments about a Queen problem yesterday
I just wanted to share that after solving different riddles, I decided to put them in my library.

One of the most popular riddles right now are River Crossing Riddles

There are currently 7 of them: https://www.problems.cc/p/HzcTyl?from=library&slug=river-crossing-riddles (the first one is obviously the Wolf, Goat and Cabbage riddle)
Probably "Missionaries and Cannibals" is my personal favourite.

Do you know any other interesting river crossing variations that I should add?

Triangle ABC is an isosceles right triangle make of paper and D is the midpoint of leg AB. If the triangle is folded so C meets D, creased, and then unfolded, what is the ratio of the two segments the hypotenuse is split into by the crease?

A professor decides to test his bright students Raj, Lisa, Ken and Lin. He shows them a bunch of oranges. 

He says,” As you can see I have these oranges and as you can count it is a Prime number less than 15. Now here is how the test will go. One by one you will pick up some oranges and leave the room. Here are the conditions to pick up the oranges. Each one must follow a separate condition. No repeating of any condition. The order of the conditions is up to you. 

1  One of you can pick up oranges that are an exact cube root of the number of oranges remaining. 

2 One can pick up oranges that are an exact square root of the number of oranges remaining. 

3 One can pick up a prime number of oranges

4 One can pick up oranges equal to the remaining students in the room. 

 At the end all the oranges must be picked up and each one of you must pick up at least one orange. 

Just to be clear, if there are X oranges in front of you and you want to use either the square root or cube root condition, then X must be either a cube or a square. And if you want to use condition 4 it must be the number of students remaining in the room. 

You can strategize of course. And each one of you must pick a separate condition. No repeats, All 4 conditions must be used. Good luck.

The students huddled up and came up with a strategy. 

Lisa : Cube root

Lin: Number of people remaining

Ken : Square root

Raj : Prime number

Then they went in a specific order. At the end all oranges were gone and interestingly each one had a different number of oranges. 

How many oranges were there? In what order did they go?  How many oranges did Lisa get?

(easy) show that a unit square can fit inside the region [0,x]^3 where x = 2 sqrt2 / 3 ≈ 0.94281 .

(bonus) show true or false: x is the minimum. i strongly believe this is true but i have no proof of it.

You are trapped inside an ancient underground library.

After hours of searching, you finally find the forbidden archive: a square stone platform at the center of a deep square pit. The manuscripts on that platform contain the only way out.

But the pit is sound-sensitive.

If anything falls into it — a pebble, a splinter, even you — an alarm triggers, and the library seals forever.

You measure the gap between your side and the platform.

3 meters.

Then you find two narrow wooden beams lying against the wall.

Each beam is 2.9 meters long.

One beam is too short.
Putting them end-to-end leaves the joint unsupported.
Dropping anything is not allowed.
Jumping is not an option.

You have two beams, a silent pit, and one chance.

How do you cross?

Santa Claus has infinitely many elves, numbered 0,1,2,3.... If each elf gives $1 to another one, is it possible that all elves receive infinite $$$ ?

[Note: this is a simplified version of the riddle "A very unbalanced directed graph"]

Give me a 10 digit whole number such that:

The first digit is the total number of zeros in the number

The second digit is the total number of nines in the number

The third digit is the total number of eights in the number

The fourth digit is the total number of sevens in the number

And so on to the 10th digit: The 10th digit being the total number of ones in the number

Possible multiple solutions

This challenge is part of the missions in the videogame Shin Megami Tensei: Strange Journey. I recreated it in a website. All info is there.

https://megami.com.uy/kti

given that the sum of 1/|u|^4 over all u ∈ Z^2 \ {(0,0)} is equal to (2/3) G π^2 .

find the sum of 1/|u|^4 over all u ∈ K , where K is the set of these 8 points tiling over Z^2 by translating 5 units in four coordinate-axis directions.

alternatively, prove that the sum is equal to (32/625) G π^2 .

^{note: i discovered a trick while trying to solve a related problem posted here awhile ago. while it is cute, it failed to work, so i repurposed it and design a new problem around it.}

In a m by n chess board, place m kings on the leftmost column. Count the number of ways to move all kings to the rightmost column, such that each tile is visited at most once.

Note: a king moves one tile to one of the 8 adjacent tile.

Find a 3x3 magic square of integers that satisfies these requirements:

1. All numbers are integers (ℤ) and different from each other.
   
2. Each row, column, and diagonal add up to 0.

Hint: check if the number at the center of the square can be 0 or not.

Example:

-1 +4 -3
-2 0 +2
+3 -4 +1

What is the smallest positive integer N such that N*pi and N*e are both within 1/1,000,000 of an integer?

Another sudoku related puzzle.

Solution:

Place ones in all the regions (the 3x3 boxes) in this order: up left, up, up right, left, down left, center, right, down, down right.

You should be able to convince yourself that no matter where in a region you place a 1, the number of options for placing a 1 in the respective region is always 9, 6, 3, 6, 3, 4, 2, 2, 1, so the number of possibilities is 9*6*3*6*3*4*2*2*1=46656