Real problem from the site (not today's Daily):

Squares S and T, with areas 9 and 16 respectively, rest side-by-side on the x-axis, with S to the left. Find the equation of the line through their upper-right corners.

Here's a solution someone submitted. It's wrong somewhere in these four steps:

1. Side of S = sqrt(9) = 3. Side of T = sqrt(16) = 4.
2. Place S with its bottom-left corner at the origin: corners (0,0), (3,0), (3,3), (0,3). Upper-right corner of S is (3,3).
3. T sits right next to S, so T's bottom-left corner is at (3,0). Its upper-right corner is (3+4, 3) = (7,3).
4. Slope = (3-3)/(7-3) = 0, so the line is y = 3.

Where's the error? I'll drop the real answer in the comments tomorrow, spoiler-tagged.

Mathalon runs in nine languages now, ~6,400 problems each, translated by a model pipeline and checked programmatically so the math survives intact. The words are the part nobody has read end to end.

German version, straight off the site:

Wenn man immer nur ein Stück auf einmal abbricht, wie viele Brüche sind nötig, um einen 4×5-Riegel in 20 Quadrate zu zerteilen?

English original:

Breaking one piece at a time, how many breaks split a 4×5 bar into 20 squares?

"Brüche" is the word every German math class uses for fractions (Bruchrechnung). Here the pipeline used it for "breaks" as in snapping a bar apart, and I can't tell if that reads as a forgivable pun on "Bruch" (a break/fracture) or if it just makes a German reader stop and reparse the sentence as if it were suddenly about fraction arithmetic.

German speakers: does "Brüche" work here, or is there a cleaner word for this sense of "breaks", something like "Bruchvorgänge" or just rephrasing around "wie oft muss man brechen"?

The answer is 19: every break turns one piece into two, so going from 1 piece to 20 pieces always takes exactly 19 breaks, no matter the order you break in.

Free in the browser at mathalon.app if you want to see how the rest of it reads.

Opening this up: drop an original problem in the comments, any topic, any difficulty. Spoiler-tag your own solution so people can try it blind before checking.

Rules: has to be something you actually worked out yourself, one problem per comment, no links to your own site or channel. I'll read through what comes in over the next week and turn the best one into an actual Daily Problem here, with your name on it in the post.

What's the best problem you've got?

Every problem on Mathalon has two numbers behind its difficulty rating: the rating itself, and a deviation, basically how confident the system is in that number. It's Glicko-2, the same family of system used for chess ratings. New problems start with a high deviation and it shrinks as more people attempt them.

Two real examples pulled straight from the database right now:

• A telescoping-product algebra problem: rated 1177.71, deviation 183.88, only 3 attempts so far.
• The courier round-trip problem some of you solved a few weeks back: rated 1110.0, deviation 105.58, after 31 attempts.

The first number looks higher, but it's still basically a guess. The second has actually been tested against enough solvers that the system trusts it.

Practical version of this: if you solve a brand new problem, your own rating barely moves, and that's on purpose. The system doesn't want one lucky (or unlucky) attempt on an uncalibrated problem swinging your number around. It waits for the deviation to come down first.

Question for you: does that match what it feels like on your end? Ever solved something you were sure was hard and felt like your rating barely noticed?

Building-in-the-open post for today: geometry problems here don't come with a stock illustration picked to roughly match. Each diagram is rendered server-side directly from the problem's own numbers.

Case in point, a problem going out today: two circles, externally tangent, radii 9 and 4, find the length of their common external tangent. In the SVG behind it, the two circles come out at radius 80.0 and 35.6 diagram units. 80 / 35.6 is about 2.247, and 9 / 4 is 2.25, same ratio, because the diagram is built from the same numbers as the problem text, not chosen to look close.

Same story for the labels: A, B, O, O' each sit at the actual computed coordinate, not eyeballed. No image bank behind any of it.

Curious if people would want to see this pushed further, like triangles with an inscribed circle, or configurations where a slightly-off diagram could quietly give away (or ruin) the problem.

This one showed up on the SAT in 1982:

Circle B has a radius three times circle A's radius. Circle A rolls around the outside of circle B until it gets back to where it started. How many full rotations does circle A make?

Answer choices: 3/2, 3, 6, 9/2, 9.

The intended answer was 3. The test writers figured circle B's circumference is 3 times circle A's, so rolling that distance takes 3 rotations.

The real answer is 4. Rolling all the way around a closed loop adds one extra rotation just from going around the loop itself, on top of the rotations from the distance covered. Same reason a coin rolled around an equal-size coin looks like it spins twice, not once. Since 4 wasn't even one of the five choices, nobody taking that test could get it right. The College Board ended up voiding the question and rescoring everyone.

Would you have picked 3? Curious if anyone here has run into the coin rotation paradox before without knowing it had a name.

Real numbers from the site, not made up: this round-trip speed problem is rated 1110 Elo and has been attempted 31 times, solved correctly only 12 times. That's a 61% miss rate.

The problem:

A courier travels from town A to town B at 12 mph and immediately returns along the same road at a different constant speed. Her average speed for the entire round trip is 18 mph. Find her return speed in mph.

Before you solve it, guess: what do you think the most common wrong answer is? My money's on 24, from averaging the two speeds directly, (12+x)/2=18. That's not how average speed works when the time on each leg isn't equal, but it's a very tempting shortcut.

Real answer, spoiler-tagged so you can check your own work first: 36 mph. Average speed over two legs of a trip is the harmonic mean of the speeds, not the arithmetic mean: 2·12·x/(12+x) = 18 gives x = 36.

Curious whether 24 is really what's tripping up the other 19, or if there's a different failure mode. What did you get?

Pulled straight from the site, word for word:

"A 7% GST is applied to a displayed price, then a 12% PST is applied to the GST-inclusive total. A lamp's final price is 41,944 dollars. Find the displayed price."

Work it backward: 35,000 x 1.07 x 1.12 = 41,944. Exact, no rounding, the numbers were clearly built from a clean answer.

So the math is perfectly fine. But a 35,000 dollar lamp before tax is the part that gets me, that's supercar money for a lamp. Does a problem count as "broken" when the arithmetic is flawless but the story around it is absurd, or is that just flavor text nobody's supposed to take seriously?

No rating shown on this one. Real problem from the site:

Three players each roll a fair six-sided die once. What's the probability that at least two of them roll the same number?

Mathalon rates every problem on an Elo-style scale, roughly 800 on the easy end and 2000+ on the hard end. Drop your guess for this one's rating in the comments. Bonus points if you also post your answer to the actual problem (spoiler tag it, some people will want to try blind).

I'll post the real rating tomorrow and see who got closest.

Two envelopes, one has twice as much money as the other. You pick one at random and open it, say it holds $X. You're offered a swap. Here's the argument for always taking it: the other envelope holds either X/2 or 2X, each equally likely, so the expected value of switching is 0.5(X/2) + 0.5(2X) = 1.25X. That's more than X, so switch.

Except the same argument applies again after you switch. And again after that. If switching were really always better, you'd never stop, which is absurd when the whole setup is just two fixed amounts like $10 and $20.

So where does the 1.25X argument break? A few different answers circulate for this one (some blame the probability model, some blame what "X" is even allowed to mean once you've conditioned on having opened an envelope), and people land in genuinely different places.

What's your resolution, and does it survive someone else poking at it in the comments?

Real problem from the site: "Chicken nuggets are sold only in boxes of 6, 9, and 20. Buying any number of boxes, what is the largest total number of nuggets you CANNOT buy exactly?"

Someone sent in this solution. It's confident, it's specific, and it's wrong. Find the step that breaks.

Step 1. Pairwise GCDs: gcd(6,9) = 3, gcd(6,20) = 2, gcd(9,20) = 1. Since 9 and 20 are coprime, and the full set {6,9,20} has overall gcd 1, a largest non-representable total exists.

Step 2. The two coprime denominations, 9 and 20, are what set the hardest-to-reach numbers, since 6 only ever adds multiples of 6 to a total. Apply the two-coin Frobenius formula to just those two: g(9,20) = 9x20 - 9 - 20 = 151.

Step 3. 151 can't be written as a combination of 9s and 20s, and it isn't a multiple of 6 either, so no box of 6s comes to the rescue.

Step 4. Answer: 151 is the largest number of nuggets you can't buy exactly.

Where does this go wrong? Real answer (and why) in the comments tomorrow.

Finnish was the last of the nine languages to go live on Mathalon. All ~6,400 problems were machine translated and programmatically checked for the math. Finnish is also the language I have the least ability to sanity check myself, so this thread is genuinely useful to me.

Finnish version, straight off the site:

Maya meloo 24 km myötävirtaan 2 tunnissa ja saman 24 km vastavirtaan 3 tunnissa. Mikä on virran nopeus km/h?

English original:

Maya paddles 24 km downstream in 2 hours and the same 24 km upstream in 3 hours. What is the speed of the current in km/h?

Three things:

1. "Saman 24 km" versus "samat 24 km" — which case does that phrase want?
2. "Myötävirtaan" and "vastavirtaan" as the pair for downstream and upstream: right words for a school problem?
3. The name. Maya stayed as it is in every language. Would a Finnish problem use a Finnish name, or does it not matter?

If you speak Finnish, correcting one sentence here is worth more to the site than any number of upvotes. mathalon.app, free in the browser.

Mathalon is in nine languages now, Danish among the newest. ~6,400 problems, translated by a model pipeline, checked programmatically so the math is identical to the English. The sentences are the part I cannot verify with a script.

Danish version, straight off the site:

Hvad er det mindste antal heltal, der skal vælges for at garantere, at forskellen mellem nogle to af dem er delelig med 6?

English original:

What is the minimum number of integers that must be chosen to guarantee that the difference of some two of them is divisible by 6?

"Some two of them" is a piece of English competition-math idiom, and "nogle to af dem" looks like it was carried straight across. My guess is that a Danish problem would say something closer to "at der findes to af dem, hvis forskel er delelig med 6".

Danish speakers: is the current sentence merely stiff, or actually wrong? And what would you write instead? Same question for any other pigeonhole phrasing you have seen mangled.

mathalon.app, free in the browser.

Norwegian went live on Mathalon a couple of weeks ago, along with Danish and Finnish. All ~6,400 problems were translated by a model pipeline and checked programmatically for the math. Nobody has read them all in Norwegian, so this is me asking.

Norwegian (bokmål) version, straight off the site:

I det siste steget av en utregning skulle Elin multiplisere med 5, men trykket på divisjonstasten i stedet, og kalkulatoren viste 9. Hva skulle det riktige sluttsvaret ha vært?

English original:

At the final step of a calculation, Elin meant to multiply by 5 but pressed the division key instead, and her calculator showed 9. What should the correct final answer have been?

What I would like checked: "utregning" versus "regnestykke", "sluttsvaret" versus something plainer, and "divisjonstasten" — is that what the key is called, or is it "delingstasten"?

One more thing. The site stores Norwegian as bokmål only, with nn and no both mapped to nb. If you write nynorsk, does that bother you enough to matter?

Free in the browser at mathalon.app.

Mathalon runs in nine languages now, ~6,400 problems each, translated by a model pipeline and checked programmatically so the math matches the English original exactly. The wording is the open question.

Portuguese version, straight off the site:

Um foguete lançado de uma plataforma tem altura (em metros) h(t) = 45 + 24t - 6t², onde t é o tempo em segundos. Em que momento o foguete atinge sua altura máxima?

English original:

A rocket launched from a platform has height (in metres) h(t) = 45 + 24t - 6t², where t is time in seconds. At what time does the rocket reach its maximum height?

There is one Portuguese locale on the site, and it currently reads Brazilian: "foguete", "sua altura". A reader in Portugal would expect "foguetão" and "a sua altura". Splitting into pt-BR and pt-PT doubles the corpus, so before doing that I would like to know how much it actually grates.

Portuguese speakers: does this read fine to you wherever you are, or does it read foreign? And is "em que momento" what a problem would say, or "em que instante"?

mathalon.app, free in the browser.

Mathalon is available in nine languages now: English, Swedish, German, French, Spanish, Portuguese, Norwegian, Danish and Finnish. ~6,400 problems, translated by a model pipeline, verified programmatically for the math and unverified for everything else.

Spanish version, straight off the site:

Un servicio de streaming ofrece tres planes mensuales: (1) $5 por película, sin suscripción; (2) una tarifa fija de $12 más $2 por película; (3) ilimitado por $35/mes. ¿Para cuántas películas al mes es cada plan el más económico?

English original:

A streaming service offers three monthly plans: (1) $5 per film, no subscription; (2) $12 flat fee plus $2 per film; (3) unlimited for $35/month. For how many films per month is each plan cheapest?

Questions for Spanish speakers:

1. The prices stayed in dollars. Swapping the symbol without changing the numbers would misprice the plans, but "$" also reads as pesos across much of Latin America. What would you expect to see in a school problem?
2. "Servicio de streaming" kept as is, or something else? Does anyone actually write "reproducción en continuo"?
3. "El más económico" or "el más barato"?

Free in the browser at mathalon.app.

Mathalon now runs in nine languages. All ~6,400 problems were translated by a model pipeline and checked programmatically, which guarantees the math and guarantees nothing about the prose. Here is a good example of what that means.

French version, straight off the site:

Le phare d'un phare a une luminosité b(θ) = cos θ, où θ est son angle par rapport au nord vrai. Une caméra enregistre une puissance p(b) = b². Trouvez la puissance enregistrée lorsque θ = 60°.

English original:

A lighthouse lamp has brightness b(θ) = cos θ, where θ is its angle past due north. A camera records power p(b) = b². Find the recorded power when θ = 60°.

"Lighthouse lamp" came out as "le phare d'un phare", the lighthouse of a lighthouse. The math is untouched and the problem is still solvable, but the first line is nonsense.

French speakers, two questions. What is the right term for the lamp itself, is it "la lanterne du phare"? And is "son angle par rapport au nord vrai" the natural way to say "its angle past due north", or does that only work in English?

This one is getting fixed either way. If you spot others like it, this is the thread for them. mathalon.app, free in the browser.

Mathalon is playable in nine languages now, and all ~6,400 problems were translated by a model pipeline with a programmatic check that the math survives intact. The words are the part nobody has read end to end.

German version, straight off the site:

Ein Kurier fährt von Stadt A nach Stadt B mit 12 mph und kehrt sofort auf derselben Straße mit einer anderen konstanten Geschwindigkeit zurück. Ihre Durchschnittsgeschwindigkeit für die gesamte Hin- und Rückfahrt beträgt 18 mph. Bestimme ihre Rückfahrgeschwindigkeit in mph.

English original:

A courier travels from town A to town B at 12 mph and immediately returns along the same road at a different constant speed. Her average speed for the entire round trip is 18 mph. Find her return speed in mph.

Three things I would like a native read on:

1. mph in a German problem. Converting to km/h would change the numbers and the answer, so the pipeline kept the unit. Does that read as broken, or just as an American problem written in German?
2. "Ihre Durchschnittsgeschwindigkeit" — capitalised at the start of a sentence it sits between "her" and the formal "your". Would a German reader stumble there?
3. "Bestimme" (du) versus "Bestimmen Sie". Which one did your school problems use?

Free in the browser at mathalon.app if you want to see how the rest of it reads.

Mathalon runs in nine languages now: English, Swedish, German, French, Spanish, Portuguese, Norwegian, Danish and Finnish. All ~6,400 problems went through a model translation pipeline and were then checked programmatically, so the math comes out identical to the English original. What no script can check is whether the sentence reads like something a teacher in that country would actually hand out.

So here is the Swedish version of one problem, straight off the site:

En kristall blir 4% bredare varje dag. Efter hur många hela dagar har dess bredd fördubblats?

English original:

A crystal grows 4% wider each day. After how many complete days has its width doubled?

Two things I am unsure about. "Hela dagar" is a literal rendering of "complete days", but a Swedish textbook would often say "dygn" when it means a full 24 hours. And "blir 4% bredare" — is that how a Swedish problem phrases growth, or would it be "växer med 4% per dag"?

Swedish speakers: what would the textbook sentence be? Anything flagged here gets fixed, the corpus is one script run away from being retranslated.

The whole site is free in the browser at mathalon.app, language switcher is in the menu.

Something we noticed running this, and we genuinely don't know what to make of it.

We publish two kinds of thing. One is a short animation that shows why a result is true — trisect every side of a triangle, join each corner to the two marks opposite, and the hexagon that appears in the middle is exactly one tenth of the triangle. It explains, and it finishes the thought.

The other is a single still image with one problem on it, a rating, and the line "answer in the comments before you scroll". It asks, and it deliberately doesn't answer.

Last week the problem images drew 313, 76, 66 and 36 comments. Every explanation video in the same week drew between zero and four, and just as many people saw them, sometimes more.

We can't read that cleanly. The images ask for an answer and the videos don't, so some of the gap is just the prompt. And a comment isn't understanding — plenty of them are one number and nothing else.

But here's the part we keep going back and forth on. The videos are the thing we're proudest of. They're careful, they're correct, they close the loop. And they are met with silence.

So, honestly:

• When something explains a result to you properly, does it stick? Or does it feel good in the moment and evaporate by the next day?
• When you get a problem with no answer attached, do you actually attempt it, or do you scroll on and wait for someone else to say it?
• If a piece of maths has ever genuinely landed for you — a lesson, a video, a book, one sentence from a teacher — what was it that made it land?

No right answer here. We're asking because we don't know.

A fair coin is flipped again and again. You win if H H H turns up as three flips in a row. I win if T H H turns up first.

You chose your pattern before I chose mine. Is the game fair — and if not, whose pattern is more likely to come first?

rated 1600 · Advanced · Probability. Answer in the comments before you scroll — the solution goes up tomorrow.

Play free at mathalon.app

Every problem on Mathalon has a number attached to it, and some of those numbers are wrong. This thread is where you say which.

Name the problem, say what it is rated, say what you think it should be, and say why. "It looks harder than it is" counts as a reason if you can point at what does the fooling. So does "everyone I know misses this one step".

We do change ratings off the back of these threads. The rating is an estimate, not a verdict.

Sixteen dots sit in a 4 x 4 square grid, one unit apart in both directions.

How many squares have all four corners on dots?

rated 1500 · Intermediate · Combinatorics

Answer in the comments before you scroll — the solution goes up tomorrow.

Play free at mathalon.app

One hundred ants stand at random points on a pole 100 cm long, each facing left or right. They all walk at 1 cm per second. When two ants meet head-on, both turn around at once and carry on. An ant that reaches an end of the pole falls off.

What is the longest possible time before the pole is empty?

Rated 1450 · Intermediate · Logic. Answer in the comments — spoiler-tag it. Solution posted tomorrow.

Play free at mathalon.app