TECHNIQUE 2 OF 4

The peg count: distinguish a target from a dead end

Each legal jump reduces the population by one. Use this invariant to check a proposed finish and to understand why a quiet board with no moves is not necessarily a win.

The rule and its limit

Starting with N pegs, reaching one requires exactly N minus 1 classic jumps. The goal checks the final peg count, not simply whether a move is available. If multiple pegs remain but no legal jump exists, the attempt has stopped short of its goal.

A common mistake

Counting a moved peg and a removed peg as two removals is incorrect: the moving peg still exists at the landing hole. A final peg outside the centre is also a valid classic finish.

PLAYABLE EXAMPLE 1

Count the work remaining

Starting with 4 pegs, how many jumps would a one-peg finish require? Find a route that actually makes all of them.

○R1C1○R1C2○R1C3○R1C4○R1C5○R2C1○R2C2○R2C3○R2C4○R2C5○R3C1○R3C2○R3C3○R3C4○R3C5○R4C1●R4C2●R4C3○R4C4●R4C5○R5C1○R5C2●R5C3○R5C4○R5C5

Loading playable study…

Select a peg, then one of its available empty landing squares. A jump removes the peg in between.

Keyboard: Tab to an enabled control, then press Enter or Space. The study starts from the diagram shown; reload can restore your own saved moves.

Try the task before inspecting the answer.

Worked moves are available when your board matches a checkpoint in the written route. If you choose another branch, inspect it, undo, or reset.

Compare the starting moves

Every jump removes exactly one peg, so a one-peg finish needs 3 jumps. Every legal first jump is evaluated. A successful continuation is verified by a complete finishing route; a no-finish conclusion exhausts all remaining legal branches. The route below is one verified finish; its last peg may be anywhere on this classic board.

First move	Immediate effect	Checked consequence
R4C2 → R4C4	Remove the jumped peg at R4C3. 3 pegs remain.	Can finish with one peg in 2 further jumps.
R4C3 → R4C1	Remove the jumped peg at R4C2. 3 pegs remain.	No one-peg finish exists after this jump; every remaining legal branch was checked.
R5C3 → R3C3	Remove the jumped peg at R4C3. 3 pegs remain.	No one-peg finish exists after this jump; every remaining legal branch was checked.

Open this study’s full route and evidence →

PLAYABLE EXAMPLE 2

A count is not a plan

Starting with 5 pegs, how many jumps would a one-peg finish require? Find a route that actually makes all of them.

○R1C1○R1C2○R1C3●R1C4○R1C5○R2C1○R2C2○R2C3○R2C4●R2C5○R3C1○R3C2●R3C3○R3C4●R3C5○R4C1○R4C2●R4C3○R4C4○R4C5○R5C1○R5C2○R5C3○R5C4○R5C5

Loading playable study…

Select a peg, then one of its available empty landing squares. A jump removes the peg in between.

Keyboard: Tab to an enabled control, then press Enter or Space. The study starts from the diagram shown; reload can restore your own saved moves.

Try the task before inspecting the answer.

Worked moves are available when your board matches a checkpoint in the written route. If you choose another branch, inspect it, undo, or reset.

Compare the starting moves

Every jump removes exactly one peg, so a one-peg finish needs 4 jumps. Every legal first jump is evaluated. A successful continuation is verified by a complete finishing route; a no-finish conclusion exhausts all remaining legal branches. The route below is one verified finish; its last peg may be anywhere on this classic board.

First move	Immediate effect	Checked consequence
R2C5 → R4C5	Remove the jumped peg at R3C5. 4 pegs remain.	No one-peg finish exists after this jump; every remaining legal branch was checked.
R3C3 → R5C3	Remove the jumped peg at R4C3. 4 pegs remain.	No one-peg finish exists after this jump; every remaining legal branch was checked.
R3C5 → R1C5	Remove the jumped peg at R2C5. 4 pegs remain.	Can finish with one peg in 3 further jumps.
R4C3 → R2C3	Remove the jumped peg at R3C3. 4 pegs remain.	Can finish with one peg in 3 further jumps.

Open this study’s full route and evidence →

No moves left can mean a loss

On a five-by-five study board, only R1C1 and R5C5 contain pegs; every other hole is empty.

The tempting move

Declare the study complete because there is no legal jump.

Why it fails

Two pegs remain. Neither has an adjacent peg to jump over, so the position cannot reach the one-peg goal.

What you can conclude

This is a dead end, not a win. Return to an earlier decision or restart the study.

Try the idea in another position

A study starts with six pegs and ends with exactly one after legal classic jumps. How many jumps occurred? Can a successful alternative use fewer jumps by choosing a better route?

Check your reasoning

Exactly five jumps occurred. Every classic jump removes one peg, so every successful route from the same six-peg start uses five jumps. The challenge is finding a route that finishes, not reducing that forced jump count.

Playfield Arcade · Updated 3 October 2026 · AI-assisted writing and code; examples checked against the local game rules. Our method · Report a correction