Can these seven pegs finish with their only peg at R3C3?
Make a prediction before opening the reasoning.
Compare the predictions
- Yes: the centre peg can be preserved.
Not supported. No one-peg centre state appears among all 65 reachable states.
- No, but one-peg finishes elsewhere exist.
Supported. The only finishing holes are R4C1 and R4C4.
- No one-peg finish exists anywhere.
Not supported. Both displayed six-jump routes reach one peg.
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Test this exact position
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Select a peg, then one of its available empty landing squares. A jump removes the peg in between.
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The position and the question
Playfield’s classic goal is one peg anywhere on the board. A centre finish is a different, stricter challenge. The centre begins occupied here, but that does not establish that the last peg can remain there.
Every legal orthogonal jump removes exactly one peg. A successful finish from seven pegs therefore takes six jumps. We will use that count to check a route, then distinguish the existence of a successful route from the set of possible finishing locations.
Show the reasoning
Play five shared setup jumps
Jump R3C2→R3C4, R4C4→R2C4, R2C5→R2C3, R5C3→R3C3, and R2C3→R4C3. Each jump has an occupied middle square and empty landing square when made. The two remaining pegs are now R4C2 and R4C3.
○R1C1○R1C2○R1C3○R1C4○R1C5○R2C1○R2C2○R2C3○R2C4○R2C5○R3C1○R3C2○R3C3○R3C4○R3C5○R4C1●R4C2●R4C3○R4C4○R4C5○R5C1○R5C2○R5C3○R5C4○R5C5After five jumps: the remaining pegs at R4C2 and R4C3 allow two final jumps. Choose either legal final jump
From those two pegs, R4C3 can jump left over R4C2 to R4C1. Alternatively, R4C2 can jump right over R4C3 to R4C4. Both are six-jump finishes with one peg. Thus at least two finishing holes are reachable, and neither is the centre.
○R1C1○R1C2○R1C3○R1C4○R1C5○R2C1○R2C2○R2C3○R2C4○R2C5○R3C1○R3C2○R3C3○R3C4○R3C5●R4C1○R4C2○R4C3○R4C4○R4C5○R5C1○R5C2○R5C3○R5C4○R5C5One valid finish: a single peg at R4C1. Ask the stronger question correctly
The two displayed routes do not by themselves prove that a centre finish is impossible. To make that claim, enumerate every legal continuation from the starting position, merging repeated occupancy states. There are 65 distinct reachable states in total. Their only one-peg states occupy R4C1 or R4C4.
Use the actual win condition
The exhaustive result rules out a centre finish on this fixed starting board. It does not make the board unsolvable under the normal one-peg-anywhere rule: the two routes already solve it. When checking a challenge, write its finish condition explicitly instead of importing a stricter goal from another peg-solitaire variant.
The tempting shortcut
The centre starts occupied, so a successful route can be arranged to leave that peg there.
Where the shortcut breaks
The current centre peg may be jumped or moved, and its presence is not an invariant. The complete reachable-state search contains no one-peg centre state.
A better decision
Choose either reachable finishing hole for the classic goal; treat a centre requirement as a separate challenge that needs its own proof.
Changed conditions
Reflect the entire starting peg position left to right, so column C becomes column 6 − C. Which one-peg finishing holes are now possible? Can the centre become a finish?
Check the changed-condition answer
Every legal jump has a reflected legal jump, and reflecting twice restores the original route. The original finishing holes R4C1 and R4C4 therefore map exactly to R4C5 and R4C2. There are no additional finishes: reflecting any supposed extra route back would contradict the original complete result. R3C3 maps to itself, so a centre finish remains impossible.
Take this back to your next game
A route proves one destination is reachable. Ruling out a destination requires evidence covering every route; here, exhaustive search supplies it.
How the claim was checked
Enumerate every reachable occupancy state and replay both six-jump routes.
- 65 distinct reachable states
- Only one-peg holes: R4C1 and R4C4
- Both supplied routes use exactly 6 legal jumps
- Horizontal reflection: only R4C2 and R4C5 finish; centre remains impossible
Read the fixed positions and supporting proof data. Computational checks establish only the claim and position stated here. They do not establish a player difficulty rating or make an unproved strategy optimal.
Playfield Arcade · Updated 4 October 2026 · Original case design, AI-assisted writing and code, with reproducible position checks. Our method · Report a correction