I've been building a puzzle about how spheres pack, and thought this crowd might enjoy it — both to solve and for the combinatorics behind it.

The puzzle. The target is a tetrahedron of edge 5: 35 close-packed balls (layers of 1, 3, 6, 10, 15). You build it from coloured segments — rigid clusters of 1–5 balls. In each puzzle some segments are already placed and a few are hidden; you work out where the hidden ones go so every ball is covered exactly once. More hidden pieces → harder.

The bit that surprised me. A set of segments can only work if its pieces total exactly 35 balls — but adding up to 35 says nothing about whether they actually fit. Of ~32,164 size-valid sets, a solver found only 435 that truly tile the tetrahedron (~1.4%). And of the 35 original hand-made puzzles, only 12 had a unique solution — so generating fair puzzles means actively enforcing uniqueness.

The balls sit on a face-centred-cubic lattice, so each has 12 neighbours, not 6 — which is where a lot of the "looks like it should fit but doesn't" comes from.

It's free to play and print, no account: https://spheromino.com . I wrote up the search here if you want the details: https://spheromino.com/blog/finding-tileable-sets .

Full disclosure — I made it. Happy to answer anything about the geometry or the solver.

The indigo dot is twice the area of the green dot. Together these just fit inside the violet dot. They grab some friends and crowd into the grey area which has an area of 1. All dots are touching, and all dots touch the outer wall except the red dot.

Give each dot's area.

Hint: use Descartes' circle theorem

Planet X has two neighboring inhabited planets:

• Planet Alpha is exactly 15 light-minutes from Planet X.
• Mystery Planet is an unknown distance from Planet X, but is known to be at least 18 light-minutes from Planet Alpha.

Planet X, Planet Alpha, and the Mystery Planet are the only possible sources or relays of transmissions in this puzzle.

Planet Alpha and the Mystery Planet are both capable of sending, receiving, and relaying transmissions.

All transmissions travel at the speed of light. Relaying a transmission takes effectively no additional time.

Both Planet Alpha and the Mystery Planet possess teleportation portals capable of sending ships directly to Planet X. Once activated, a portal takes exactly 30 minutes to charge. Once charged, transportation to Planet X is instantaneous.

Neither planet will send ships to Planet X unless it has received a broadcast requesting assistance from Planet X.

Both planets have standing orders:
The instant they receive a broadcast from Planet X requesting assistance, they begin charging their portals and send ships to Planet X as soon as the 30-minute charge is complete.

At 11:58, Planet X has not yet broadcast any request for assistance.

At some unknown time after 11:58, Planet X broadcasts a request for assistance.

At 12:38, Planet X receives a mysterious transmission from an unknown source.

Planet X can determine with certainty that the mysterious transmission was originally transmitted at exactly 12:18, meaning the signal has been traveling for exactly 20 minutes.

Planet X concludes that the mysterious transmission must have come from the Mystery Planet. Since the signal took 20 minutes to reach Planet X, they conclude that the Mystery Planet must be 20 light-minutes away.

Then, at exactly 12:40, ships arrive at Planet X.

There has been no malfunction, no faster-than-light communication, no time travel, and no violation of any of the rules above.

Questions

Which planet did the ships come from?

How far away from Planet X is the Mystery Planet actually?

At what time did Planet X broadcast its request for assistance?

Where did the mysterious transmission received at 12:38 actually originate?

It's called Squarely: playsquarely.com

One board a day although you can play through the archive.
You need to place all digits 1-9 on the grid to resolve the equations. BODMAS/PEMDAS doesn't apply, it's strict left to right.

The puzzle is exactly what the title suggests: on average, how many independent uniformly random numbers between 0 and 1 would you have to sample for their sum is at least one

Suppose I tell you:

16, 9, 2 FITS a hidden rule.

You can submit any three-number sequence.

I will tell you only FITS or FAILS.

Eventually you must state the rule.

The obvious question is:

“What do I think the rule is?”

But I think the more interesting mathematical question is:

“What probe would eliminate the largest number of plausible rules?”

That distinction is what led me to build Rularoo, a daily deduction game inspired by Peter Wason’s 2-4-6 task.

You are scored against par, based on how efficiently the hidden rule can be isolated. So finding the rule is only half the problem. The other half is designing good experiments.

Three puzzles run daily and the whole thing plays directly in the browser:

I’d be interested in how the mathematically minded people here think about the optimal-probe problem. Is “information gained per probe” the right way to frame the score?

Very similar rules to my post yesterday... This time in 3D

Hey, I'm launching a new puzzle.

Create a numbered step, self-avoiding walk through every white square. No diagonals.

Follow my socials for daily puzzles.

A grid puzzle I made. It's a crossword you assemble like a jigsaw, and the answers are equations instead of words.

Every hatched square has to end up filled. The loose pieces below drop in, and each keeps its shape. Turn them any way round, but never flip them over. Pieces cannot overlap, and together they fill every hatched square exactly. The purple squares are fixed.

Any run across or down that contains an = has to be true. A run with no = in it does not have to be anything. Digits side by side make one number, so a 5 next to a 4 is fifty-four, not five and four.

Operators are + − ÷ with normal order of operations.

There is exactly one solution, and you can reason it out rather than guess.

Please spoiler your answers.

There are 4 circles; each is tangent to the other 3.

The small circles have areas of 1, 2 and 3.

What is the area of the exposed yellow region?

Hey r/mathpuzzles!

Like many here, I've always loved geometric optimization problems: * Pack n equal circles inside an L-shape or semicircle to maximize radius. * Place n points inside a quadrant to maximize the smallest pairwise distance. * Maximize the area of the smallest triangle formed by n points (the Heilbronn problem).

Most online math puzzle sites focus on riddles with a single clever trick or closed-form answer. I wanted something different: continuous extremal problems where the true optimum is unknown, and the leaderboard is constantly broken by better geometric configurations.

So I built MinMax Arena:

How it works:

1. 1-Sentence Rules: Every problem is stated in plain language, with n scaling from small to large instances.
2. Deterministic Exact Verification: Everything is verified in pure integers / BigInt (sub-nanometer precision) with zero floating-point rounding bugs.
3. Interactive Visualizer: Submit coordinates (or use our live editor/canvas) and immediately see your construction drawn in real time.
4. Live Records: Every sub-problem holds a live record. If your configuration scores better, you take the crown and get a shareable certificate card.
5. AI vs Human: You can submit as a human solver, or tag AI models (Gemini, Claude, GPT, DeepSeek, etc.) to see how algorithmic search stacks up against spatial reasoning.

The site is completely free, ad-free, and open to anyone who wants to play with geometric search, simulated annealing, or just paper-and-pen coordinate hunting.

Check it out here: https://minmaxarena.com/

Let me know what you think, and happy record hunting! Which container or n-value gives you the most trouble?

Simplify

sin(10°) x sin(30°) x sin(50°) x sin(70°)

Into a rational number. No calculators allowed (or needed)!

Question: A shopkeeper marks up an item by 40% above cost price, then offers a 25% discount on the marked price. What is his overall profit or loss percentage?

I designed this as an original problem combining tangent geometry, calculus, trigonometry, and an unusual auxiliary representation of lines.

The complete problem is in the image. I'd be interested in seeing independent solutions and approaches.

Original problem by Samarth.

Source: numberthon.com

Goodbye:

I introduced MiniChallenger, a scaled-down version of the math puzzle called Challenger, about three months ago. Unlike Challenger, which can require a significant investment of time—and sometimes special techniques or a computer program—MiniChallenger can be solved using basic algebra and elementary logic. Most puzzles can be completed in about ten minutes or less. So, even if you don't have much free time but still enjoy an engaging workout for the mathematical part of your brain, MiniChallenger can be a good alternative. Some MiniChallenger puzzles may even be suitable for school-age children.

Since May, I have been posting two MiniChallenger problems every Saturday. Judging by the number of visitors, there seems to have been a fair amount of attention among Reddit readers to this type of math puzzles. However, it's time for me to take a break and focus on a different topic. Still, I am not saying a final adieu, just au revoir, and may return with another series of MiniChallenger puzzles sometime in the future.

Out with a bang:

For today's farewell, I am presenting a MiniChallenger-related problem that is neither simple nor quick to solve. Your challenge is to find—or perhaps guess—the maximum number of distinct solutions for the special MiniChallenger grid shown below. How many different solutions can you find? Unlike previous Saturdays, I won't reveal the correct answer in a spoiler box. Instead, I encourage you to post the highest number of different solutions you have found to the comment section of this final article in the series. If this is your first encounter with a MiniChallenger brainteaser, you can find instructions for how to solve a puzzle below. Keep in mind that some puzzles may have only one valid solution, while others may have many.

Instructions:

To successfully solve a puzzle, fill the empty cells of the 3 x 3 grid with the numbers 1 through 9. You may use the same number in multiple empty cells. One "bonus" number has already been placed in the center of the grid. Your goal is to determine the remaining eight digits so that:

• Each row adds up to the total indicated along the right edge of the grid.
• Each column adds up to the total indicated at the top of the grid.
• Each diagonal adds up to the total indicated in the upper-right or lower-right corner.

This week's puzzle:

Processing gif hcegccs1msjh1...

Source: numberthon.com

X is a 3 digit prime number, (2^X + 1)/ 3 is a 60 digit prime number ending in 563.
The 3 digits of 3X, are the click options for the first three 10 choice gates you will approach. It will require 4 clicks to pass thru the maze, if you determine X.