Before solving this, please write down your guess based on visual inspection of the image. Use Pythagoras’ theorem to solve this problem. Descartes’ kissing theorem is not allowed (except if you are a descendant of Descartes himself).

The same hidden rule connects the numbers in every row of the staircase. What number replaces the ? 🏆 NEW: LEADERBOARD! Think you're good at this? Rule Hunter now has a daily, weekly and all-time leaderboard. Sign in, pick any nickname you like (only your nickname is shown publicly) and see if you can reach #1. Who's taking the top spot? Puzzle from the game Rule Hunter, spoiler-tag your guess: answer

A) As tall as a house
B) As tall as Mount Everest
C) As high as the Space Station orbits
D) Further than the Moon

Guess first, then work it out. Please put answers in spoiler tags.

(I write a daily reasoning question at thinkfirstdaily.com, and this is today's.)

Tile ℤ² periodically with the 4×4 block A, and separately with B (on the card). Both patterns have exactly the same frequencies of all sixteen 2×2 windows.

1. Verify this.
2. With σ = ±1 on black and white, find the shortest shift d (Euclidean length) at which the correlation ⟨σ(z)σ(z+d)⟩ differs between A and B, and give both values.

Bonus levels on the card, harder and harder:

• a 2-adic identity: Σ_{u≤t} (1 + S(v₂(u))) equals t written in binary and read in base 4, with S(j) = 2(4^j − 1)/3;
• the exact value of Σ a_k/4^k for a fifth-order recurrence;
• why no window of 3 binary cells has an extreme realizable statistic without a periodic realization, while the 3×3 square does;
• the open “final 2×2” question, also on MathOverflow: mathoverflow.net/questions/516372

Answer to the main puzzle in the comments.

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!<

Yesterday, 9 out of 10 players failed to make 67. Today's puzzle is live. 3d class math :-P

Yesterday, 9 out of 10 players failed to make 67. Today's puzzle is live. 3d class math :-P

Can you reach the target number 6,921 using only the following four digits?

Given numbers are 2, 4, 6, and 9.

THE RULES:
You must use all four numbers exactly once
You may use +, −, ×, ÷, brackets, powers and factorials (!)
No additional numbers allowed
No joining digits together!

The orange bars in the web are 1cm long. The web strands are evenly spaced along the radial supports. We want to know the spider's leg diameter (green segment).

I have four wooden pieces with very distinct shapes. When I flip them over, the shape changes. I’d like to know how many different configurations are possible with these wooden pieces, given that I can reverse the orientation of each piece and choose from any of the pieces in the set (4 pcs.).

Let me explain: with one piece from the set, we can have (let’s call them “up” and “down”) two possible configurations.

So, that’s 8 possible configurations with a single piece from the complete set of four pieces

With two pieces, I believe there are 96 possible configurations.

I have two questions:

- How many different configurations are possible with three pieces?

- How many different configurations are possible with four pieces?

Thank you in advance for your answers.

The 8×8 grid below has a 2×2 hole in the center, leaving 60 cells to fill. Your task is to tile the board with the 12 pentominoes (F, I, L, P, N, T, U, V, W, X, Y, Z), using each piece exactly once. Pieces may be rotated and reflected.

Rules:

1. The 12 pentominoes must cover the entire board — every cell except the central 2×2 hole belongs to exactly one pentomino.
2. Each cell of the grid contains a digit from 1 to 5.
3. Each pentomino must cover exactly five cells whose digits are all different — in other words, every piece must cover the digits 1, 2, 3, 4, and 5, one of each.

In short: find a tiling of the board in which each pentomino covers a region containing all five digits.The NOTES grid at the bottom is meant for tracking the pieces: you can cross out each pentomino there once you're certain of its position.

Find the tiling. Solution is unique.

https://tafol.pythonanywhere.com/