Problem
Clues that reduce trial and error
The only move is to reverse a block that reaches the right-hand end. Start from 653412. Use exactly four moves. Reach 123456.
Reasoning · Grade 4-2 Games of Wit
Your answer
Watch out Reversing the middle is tempting, but a block must always run to the right end.
How to solve
Strategy Work Backwards — Each move offers 5 different block lengths, so blind trial and error would mean checking 5 x 5 x 5 x 5 = 625 four-move sequences - far too many. Instead I look at the two ends of the journey for a clue. The finish 123456 is the exact mirror image of 654321, and turning the whole number around is one legal move, so if I can reach 654321 in three moves the fourth move is free. That single observation replaces the search with a much easier goal, and from there I only have to move the 4 into place, which takes two moves, and tidy the tail, which takes one.
1STEP 1
Be exact about what a move does
The reversed block always includes the right end.
4213 → 4312, 4312 → 4321, 4321 → 1234
Testing the rule on the worked Example first is the cheapest possible insurance - if my reading of the move reproduces all three steps of the Example, it is right.
4.NBT.A.2Make A Systematic List2STEP 2
Work backwards from the finish to find a target
Working backwards, the target is 654321.
654321 → 123456 (turn around all six)
Looking at the last step first is much easier than looking at the first step, because there is only one arrangement that is the mirror image of the finish.
4.OA.C.5Work BackwardsWorking backwards from the finished order names a target that the last move must have come from.
Why?
Reversing a tail undoes itself, so running a move backwards from the finish is as legal as running it forwards.
🧱Inverse operationsSubtraction undoes addition and division undoes multiplication — they reverse each other.
Why?
Only a few arrangements could reach the finish in one move, so naming them cuts the search down to a short list.
🧱Rule out the rest and the last one is forcedIf every possibility but one has been ruled out, the survivor must be the answer — no further checking is needed.
3STEP 3
Move 1: send the 4 to the right-hand end
Send the 4 to the end: 653214.
653412 → 653214
Since only the right-hand end can ever be touched, a digit stuck in the middle has to be parked at that end first - which is a rule a young solver can discover just by trying one move.
4.NBT.A.2Guess And Check4STEP 4
Move 2: bring the 4 next to the 5
Bring the 4 beside the 5: 654123.
653214 → 654123
A digit sitting at the right-hand end can be dropped into any earlier position simply by choosing how long a block to turn around, which makes this the natural way to place a digit exactly.
4.NBT.A.2Guess And Check5STEP 5
Move 3: reverse the tail to finish 654321
Reversing the tail gives 654321.
654123 → 654321
When the only thing wrong is a tail written in the opposite order, one turn-around of that tail fixes it in a single move.
4.OA.C.5Work Backwards6STEP 6
Move 4: turn the whole number around
Reversing all six gives 123456.
654321 → 123456
The number that runs 6, 5, 4, 3, 2, 1 downwards is the mirror image of the one that runs 1, 2, 3, 4, 5, 6 upwards, so a single reversal turns one into the other.
4.OA.C.5Work Backwards7STEP 7
Replay the four moves as a check
That is exactly four moves.
653412 → 653214 → 654123 → 654321 → 123456
Replaying the whole chain and checking the two rules - block reaches the end, digits stay 1 to 6 - catches any slip in a few seconds.
4.NBT.A.2Make A Systematic ListAnswer
653214, 654123, 654321
653412 → 653214 → 654123 → 654321 → 123456
Each of the three answers is a 6-digit arrangement using 1, 2, 3, 4, 5 and 6 exactly once, which every arrangement in this puzzle must be, since turning a block around only reorders digits and never changes which ones are present. Each step really is a legal move: 412 to 214 is the last three, 3214 to 4123 is the last four, 123 to 321 is the last three, and 654321 to 123456 is all six - all four blocks reach the right-hand end. The count is right as well, with exactly four arrows and three boxes. Four moves also feels like the right amount of work: at the start only the 6 and the 5 are in the correct relative order for 654321, so at least the 4, the 3, the 2 and the 1 all have to be repositioned.
Takeaway
Look at the finish first: 123456 is just 654321 flipped, so aim for 654321 and the last move comes free.
- Be exact about what a move does
- Work backwards from the finish to find a target
- Move 1: send the 4 to the right-hand end
- Move 2: bring the 4 next to the 5
- Move 3: reverse the tail to finish 654321
- Move 4: turn the whole number around
- Replay the four moves as a check