Proof cases
A real case · you decide
Theo and the line he did not prove
Theo — a 16-year-old who wants to prove everything, including where the proof starts
Line 0 of every Proof Sweeper board reads “Given: the opening is safe.” Theo asks why that line needs no rule.
Every later line cites a rule and some numbers. Those numbers only exist because the opening was revealed.
Something has to come first, or there is nothing to cite.
What is line 0?
Every argument rests on something. Saying out loud what it rests on is the start of being rigorous.
A real case · you decide
Amara cites the done rule
Amara — a 15-year-old writing her first accepted proof line
Amara wants to open C4. Next to it is a 1 at B3, and B3 already touches one proven mine.
The board asks for three things: the claim (safe), the rule (done) and the premise (B3=1).
If any of the three is wrong, nothing opens.
What does the accepted line say?
Writing the reason down feels slow at first. It is also what makes the step checkable by someone else.
A real case · you decide
Felix and the only places left
Felix — a 17-year-old who can see mines but struggles to justify them
A 3 at E5 has three hidden neighbours left and no proven mines yet. Felix knows all three are mines.
He writes “E6 is a mine — by full (E5=3)” and it is accepted.
He tries the same citation for a fourth square, F7, which touches a different number.
Why is the F7 line rejected?
A lot of weak arguments use a true fact to support a claim the fact does not reach.
A real case · you decide
Yuki and the order of the premises
Yuki — a 16-year-old stuck on the first board that needs the subset rule
Yuki sees that the hidden squares of a 1 at D2 are all among the hidden squares of a 2 at D3.
She submits “subset (D3=2, D2=1)” and the board explains that the smaller set comes first.
The rule reads: the squares the 1 touches are inside the squares the 2 touches.
How should the citation be written?
In logic, the direction of a relation matters as much as the relation.
A real case · you decide
Omar and the flag he only believed
Omar — a 17-year-old who flags by instinct in ordinary Minesweeper
Omar is sure G2 is a mine. He wants to use it to prove its neighbours safe with the done rule.
His citation is rejected: G2 is not a proven mine yet, so the number next to it still has a mine unaccounted for.
He is probably right about G2. Probably is not enough for a proof.
What should Omar do?
Proofs are slower than hunches because each step can be trusted by someone who does not trust you.
A real case · you decide
Beatriz and the rejection that helped
Beatriz — a 15-year-old who hears “wrong” as “you are bad at this”
Beatriz’s step is rejected. The message says the 2 at C5 still has two hidden squares that could hold its missing mine.
Her first instinct is to close the tab.
Then she reads it again: it tells her exactly which fact is missing.
What is the most useful way to read the message?
Mathematicians spend most of their time on proofs that do not work yet. That is what the work looks like.
A real case · you decide
Sven and the board you cannot lose
Sven — a 16-year-old who tries to trick the board into opening a mine
Sven keeps trying citations that are nearly right, hoping one slips through and opens a mine.
Every one is rejected, and nothing on the board changes when a step is rejected.
The checker never looks at where the mines really are. It only checks whether the rule and the numbers force the claim.
Why can a Proof Sweeper game never be lost?
That is what “proof” means: not “I am confident”, but “this cannot be otherwise.”
A real case · you decide
Zara and the level that measured itself
Zara — a 17-year-old designing puzzles for her school’s maths club
Zara wants a board that forces people to use the subset rule. Her hand-made boards keep turning out easy.
Proof Sweeper builds a board, then solves it with a perfect logician and counts how many subset steps were needed.
A board that needs none is thrown away for the Proof tier. The Hard tier needs at least three.
How should Zara check her own boards?
Designing a good test is itself a kind of proof: you show the test requires what it claims to test.
A real case · you decide
Nikhil and the numbered lines
Nikhil — a 16-year-old whose written maths loses marks for missing steps
When Nikhil clears a board, Proof Sweeper shows his proof: twenty numbered lines, each citing a rule and numbers.
Line 12 uses a mine proven on line 9. Line 9 used a square opened on line 4.
Nothing on line 12 would make sense without the lines above it.
What makes the written proof trustworthy?
The “missing steps” on a maths test are missing links in this chain. Now Nikhil can see where the links go.
When two ideas meet
A real case · you decide
Amara and Omar: prove the mine, then cite it
Amara & Omar — a first-time prover whose done step keeps being rejected, and a friend who never uses a hunch as a premise
Amara writes “C4 is safe — by done (B3=1).” The board rejects it. B3 does touch a mine, she is sure of it.
Omar asks where that mine was proven. It was not: Amara can see it, but no line says so.
A 2 elsewhere has exactly two hidden neighbours, one of which is B3’s mine.
What should Amara write first?
Every step stands on the steps before it. Omar’s habit is what makes Amara’s step legal.
A real case · you decide
Nikhil and Felix: the link that does not connect
Nikhil & Felix — a student who sees a proof as a chain, and a friend who learned where each premise’s reach ends
Nikhil reads a friend’s proof. Line 6 says “E7 is safe — by done (B3=1)”. Every line above it was accepted.
Felix counts: E7 is three columns away from B3. It is not one of B3’s neighbours.
So line 6 is not checkable from the lines above, however good they are.
What is wrong with line 6?
Reading a proof is checking every link. Writing one is making each link checkable.