8th Grade · Ages 13–14
The Counting-Information Year
Thirteen is old enough for the most useful habit an adult has: proving a thing is possible before spending a week looking for how.
- 1 Hard Thing
- 5 days · 120 minutes
- 3 unmarkable questions
The Hard Thing · about a grade above
Twelve Coins
Twelve coins, one fake. The fake is either heavier or lighter than the rest and you do not know which. You have a balance scale and three weighings. Find the fake — and say whether it is heavy or light.
Stuck for ten minutes? Good. Open one hint, not three.
1Hint 1
Before you try anything: three weighings, three outcomes each. How many different answers can the scale give you in total?
2Hint 2
There are 24 possible truths — twelve coins, each either heavy or light — and 27 possible outcome sequences. It fits, but barely. That tells you how tight the solution has to be.
3Hint 3
Weigh 4 against 4, and label everything you learn — including what you learn about the four you left off.
The answer — only after you have written something down
Weigh 1-4 against 5-8. If they balance, the fake is in 9-12, and two weighings against known-good coins settles it. If 1-4 goes down, then one of 1-4 is heavy or one of 5-8 is light; now weigh 1,2,5 against 3,4,6 — whichever way it tips leaves at most three suspects with a known direction, and the third weighing separates them. Draw the full tree yourself. The point is not the tree: it is that the count told you a tree existed before you found one.
The genius move
“Count the possible answers and the possible outcomes first. If the outcomes cannot cover the answers, stop — no amount of cleverness will save it.”
Five days, five muscles
The week
Never the same muscle two days running. The weekend is not on this list on purpose — being bored is the sixth muscle and it needs two clear days.
- MondayThe Hard Thing
Twelve Coins. Do the 27-versus-24 count before touching the coins.
30 min - TuesdayBuild
Draw the whole decision tree on one sheet. It should fill the sheet.
30 min - WednesdayRead up
Find out what a bit is, and why one yes/no question carries exactly one of them.
20 min - ThursdayTeach
Teach only the counting argument. Forty seconds, and it is the real content.
15 min - FridayBreak
Thirteen coins is 26 truths, still under 27. So does 13 work in three weighings? Find exactly where it breaks — the edge case is the interesting part.
25 min
Start Monday now
Read the problem once more, out loud, and press start. Solving it is not the measurement.
Nothing rung today yet
The twenty-five minutes
Four phases. The middle one is most of it, and it is meant to feel slow.
- 1SitRead the problem again, out loud. Write nothing yet.
- 2AttackTry the wrong thing. It is faster than thinking about the right one.
- 3StuckWrite down the exact place you are stuck. “I don’t get it” is not a place.
- 4LogOne sentence: what you will try tomorrow. Then close the book.
No answer key
Three questions you cannot mark
Ask one at dinner. Do not resolve it. A question that survives the washing-up is doing its job.
- ?1
How much does one question weigh?
- ?2
Is there information in a result you already expected?
- ?3
Could you design a question that halves the possibilities every single time?
Where this goes
The ladder
The same idea, three times, roughly a year apart. This is what depth looks like from the outside.
This year
Twelve coins, three weighings, direction unknown.
Next year
The general rule: n weighings handle (3ⁿ − 3)/2 coins. Derive it rather than look it up.
The year after
Information theory — and the discovery that this puzzle and file compression are the same puzzle.
What lives on the desk this year
- A sheet large enough for the whole tree
- Coins and a balance
- A ruler, for drawing branches straight
- A folder of proofs you wrote yourself
- A question you have been carrying for months
The rest of 8th Grade
One hard thing a day is the sharp end. It sits on top of an ordinary year.
Last updated: September 2026