Problem
Mirror the opponent to keep piles equal
One plate holds 6 marbles and the other 4. On your turn you take at least 1 from a single plate. You may never take from both plates in one turn. Find how to be the one who takes the last marble.
Reasoning · Grade 4-2 Nim Games
Your answer
Watch out Grabbing a lot looks strong, but the key is making the plates equal.
How to solve
Strategy Work Backwards — The winning move at the start is invisible, but the end of the game is not: taking the last marble wins. So I work backwards from the finish to find which positions are hopeless for whoever has to move. To keep it easy I first test tiny positions like 1 and 1, or 2 and 2, with real counters, then turn what I notice into a rule for 6 and 4.
1STEP 1
Work backwards from the end of the game
Reason backwards from the end.
Thinking about the last move first is exactly how you plan the end of a board game — you decide what you want the table to look like, then aim for it.
4.OA.A.3Work Backwards2STEP 2
Test the smallest positions with counters
Small cases show equal plates lose for the player to move.
(4,4) → (3,3) → (2,2) → (1,1) → (0,0)
Moving real counters makes the pattern visible: equal plates are bad for whoever has to move, unequal plates are good.
3.OA.D.9Create A Physical Representation3STEP 3
State the rule the small cases reveal
So the goal is to hand the opponent equal plates.
One rule covers every position at once, so you never have to memorise a list of good and bad plate counts.
3.OA.D.9Solve An Easier Related Problem4STEP 4
Make the first move: even the plates up
Go first and take 2 from the plate of 6.
6 - 4 = 2 → (6,4) → (4,4)
Finding how many to take is one subtraction: the difference between the two plates.
4.OA.A.3Work Backwards5STEP 5
Then copy the opponent for the rest of the game
After that copy the opponent exactly.
(4,4) → (4,1) → (1,1) → (0,1) → (0,0)
You never have to plan ahead — the opponent's move tells you your move.
4.OA.A.3Solve An Easier Related ProblemOnce the two plates are level, copying the opponent's move on the other plate always leaves you with a move to make.
Why?
Every move the opponent makes has exactly one mirror move available to you, so you can never be the one who runs out.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
Copying keeps the plates level after each of your turns, so the position handed back to the opponent is always the balanced kind.
🧱Even and odd behave predictablyTwice any whole number is even and adding an odd amount always swaps even with odd, so a total's parity is settled by its parts before anything is computed.
6STEP 6
See why you must be the one to take the last marble
A copy always exists, so the last marble is yours.
Because you always leave the plates matching, the opponent can never leave them both empty, and someone has to take the last marble.
3.OA.D.9Work BackwardsAnswer
go first, take 2 from the plate of 6, then copy
6 − 2 = 4
The rule has to work for every reply, so check the extremes. If the opponent empties a plate of 4, you empty the other plate of 4 and take the last marble — you win. If the opponent takes just 1, you take 1 and hand back 3 and 3, still equal. Both plates being equal after each of your turns means they can never be emptied by the opponent's single-plate move. Checking every position with plates up to 8 by machine confirms it: the losing positions for the player to move are exactly the equal ones, and from 6 and 4 the only winning first move is taking 2 from the plate of 6.
Takeaway
Even the two plates up on your first move, then copy your opponent every turn — a mirror never loses.
- Work backwards from the end of the game
- Test the smallest positions with counters
- State the rule the small cases reveal
- Make the first move: even the plates up
- Then copy the opponent for the rest of the game
- See why you must be the one to take the last marble
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