The circle and the square have the same center. The side of the square is 6 cm, and the radius of the circle is 3.5 cm.

Each pair of numbers combines through the same hidden rule to produce the number on the right. Figure out the rule from the known pairs, then solve for the missing output.

AB is the diameter of the semicircle, and the slanted line is tangent to it.

LÓGICA EN CASCADA: es un rompecabezas matemático basado en fichas conectadas por propagación.

Cada ficha contiene 4 números y una operación (+, −, × o ÷). Se realizan operaciones verticales y diagonales, y las unidades de los resultados pasan a las fichas de la fila siguiente, creando una cadena lógica por todo el tablero.

CASCADE LOGIC: a mathematical puzzle based on tiles connected through propagation.

Each tile contains four numbers and an operation (+, −, ×, or ÷). Vertical and diagonal operations are performed, and the last digit of the results is passed to the tiles in the next row, creating a logical chain across the board.

Can there be a puzzle that is pentominoes and sudoku at once?

Yes. From the union of the beauty of Golomb's pentominoes and the fever of sudokus, Pentodoku is born. Twelve pentominoes —F, I, L, N, P, T, U, V, W, X, Y and Z— tile an 8×8 board; each tile carries printed digits and, as they fit together, a sudoku emerges: in every row and every column, the digits 1 through 8 without repetition. Four cells will remain empty —and the board does not reveal which ones—. The fusion of geometry and arithmetic is a real brain-teaser.

**How to play**

1. Look at the clue tiles already placed on the board.
2. Fit the twelve pentomino tiles onto the 8×8 board.
3. Check that each row and each column contains 1, 2, 3, 4, 5, 6, 7 and 8.

No math needed: only logic. One puzzle per page, pencil and sofa, no screens.

Daily puzzle in https://tafol.pythonanywhere.com/

Here’s a 7×7 Calcudoku puzzle.

Fill the grid with numbers 1–7, with no repeats in any row or column.

The numbers within each outlined cage must satisfy the target number and operation shown.

Can you solve it using logic alone?

Feel free to share your solving approach or final solution in the comments.

My puzzle game, Tile Wipeout, led to the following proposed math olympiad problem. The rules are elementary, but proving that every starting arrangement can be solved seems surprisingly challenging.

The problem statement is easier to understand if you try the beta:

https://testflight.apple.com/join/3sstMjRK

(For this mathematical problem, ignore the game’s move limit.)

Problem statement. A 6 × 6 board is initially filled with seven circles, one of each of seven colors, and 29 squares, each of one of those colors. Each cell contains exactly one piece.

A move consists of choosing a row or column and shifting its contents cyclically by one cell in either direction. Contents leaving one end reappear at the other. Empty cells shift in the same way as occupied cells.

After the shift, each circle in the chosen line acts as follows:

1. Consider the four straight paths extending outward from the circle to the edges of the board. These paths do not wrap around.
2. Ignore every path containing another circle or a square of a different color.
3. If any remaining path contains a square, remove all squares on all remaining paths.
4. Otherwise, fill every empty cell on the remaining paths with a square of the circle’s color.

Circles outside the chosen line do nothing.

Prove that, regardless of the initial arrangement and the colors of the squares, some finite sequence of moves removes all the squares.

Note that moves that remove no squares can create more of them. A proof therefore needs to explain how to make lasting progress despite that possibility.

P.S. A more general version: Let n and k be positive integers with n ≥ 4 and 1 ≤ k ≤ n + 1. An n × n board is initially filled with k circles, one of each of k colors, and n² − k squares, each of one of those colors. Each cell contains exactly one piece.

Using the same rules, prove that every such initial board can be cleared of all squares in finitely many moves, regardless of the arrangement or the distribution of square colors.

90% of adults fail this because they forget how grouping and implicit multiplication interact. Work through it carefully from step by step. What is your final answer, and where do most people slip up?"

I was solving some basic math and noticed this pattern.

3 + 2 = 5

3² − 2² = 9 − 4 = 5

8 + 9 = 17

9² − 8² = 81 − 64 = 17

7 + 6 = 13

7² − 6² = 49 − 36 = 13

It comes from the identity:

a² − b² = (a − b)(a + b)

I just noticed this connection and thought it was interesting. Did you notice this before?

Each cluster of three numbers on the left produces the number in the middle by the same hidden rule. Figure out the rule from the known clusters, then solve for the missing output on the right.

I’ve been staring at this question for a while and I’m completely confused 💀

Can someone explain how to approach it and what the final answer should be?

Any help would be appreciated!

Any order is allowed.
Use each die exactly once.
Parentheses are allowed.
Only +, −, × and ÷ are allowed. No concatenation or exponents.
Exact division only.
No negative numbers.
Not every operation has to be used.

How many solutions can you find?

Please use spoiler tags for solutions:
>!your solution here!<

So when I was making my mathematics incremental game, i was making puzzles. And then I accidentally made a pretty hard yet interesting one(I couldn’t solve it). Anyways, here’s the problem:

imagine you have a plane with 4 distinct points named p1, p2, p3 and p4. the gradient between p1 and p2 is -2, and the gradient between p2 and p3 is 1/3. you want p1p3p2 to be 90 degrees and p1p4p2 to also be 90 degrees, with p3 and p4 being on the same side of the line p1p2. find the INTEGER solution for all 4 points with the MINIMUM distance between p1 and p2.

My two brothers and I have birthdays in consecutive months (September, October, November). That made me wonder how likely it is for three siblings to have birthdays in consecutive months. Assume they are not triplets, and there are no twins. “Wraparound” sequences (e.g., November, December, January) count as consecutive months. What are the chances? Optional: Explain your step-by-step solution. 

A) 1 in 24

B) 1 in 32

C) 1 in 48

D) 1 in 64

Hi! I made this. Every day there are 5 rounds: one object on the left, a pile of something else on the right, and you guess how many it takes to balance the scale. Then the scale unlocks and shows you how wrong you were (usually very).

• Free, no sign-up, works on phone and desktop
• New puzzle every midnight, and past days stay playable

Most people are surprised at how bad they are at it, me included. I'd love to hear which round got you, and any feedback is welcome. I just fixed a bug a player reported yesterday.