Conway’s Soldiers: The Peg-Jumping Puzzle No Army Can Advance Past Row Four
The Puzzle With an Invisible Ceiling
Conway’s Soldiers is a peg-jumping puzzle played on an unlimited square grid. Soldiers may jump horizontally or vertically over neighboring pieces, removing each piece they cross. Although players may begin with any finite army below a boundary line, no legal sequence of moves can place a soldier beyond the fourth row above it.
That result feels impossible in its own right. The board has no side walls, there is no stated limit on the size of your army, and soldiers may move in every horizontal and vertical direction. Surely a sufficiently large force could build a path to row five.
It cannot—and mathematician John Horton Conway found an elegant way to prove it.
How Conway’s Soldiers Works
Draw a horizontal boundary across a square grid. Every square above the line begins empty, while you may arrange a finite number of counters, pegs, coins, or other “soldiers” on squares below it.
A legal move follows three rules:
- One soldier jumps over an adjacent soldier.
- It lands on the empty square immediately beyond that soldier.
- The jumped soldier is removed.
Jumps may go up, down, left, or right, but not diagonally. This makes the game a close relative of peg solitaire.
Your goal is to send one soldier as far above the boundary as possible. The first empty row above the line is row one, the next is row two, and so on.
The early targets are achievable with surprisingly small armies:
- Row one requires at least 2 soldiers.
- Row two requires at least 4.
- Row three requires at least 8.
- Row four requires at least 20.
- Row five cannot be reached by any finite army.
These minimum totals are recorded in references such as Wolfram MathWorld’s Conway’s Soldiers overview.
Why More Soldiers Do Not Guarantee More Progress
Every jump costs one soldier. If two pieces participate in a jump, only the jumping piece remains afterward.
That means an upward advance needs support. To move a soldier higher, another soldier must already be waiting in exactly the right position. Creating that supporting soldier may require several earlier jumps, each of which removes another piece.
The army therefore behaves less like a crowd climbing a hill and more like a rocket burning fuel. Pieces near the boundary are sacrificed to concentrate the army’s remaining power into fewer soldiers farther forward.
Reaching row four is difficult but possible. A carefully designed formation of 20 soldiers can do it. Reaching row five, however, is not merely more difficult. It crosses a mathematical boundary between hard and impossible.
This is similar to the distinction explored in the 15-Puzzle’s famous impossible challenge. In both puzzles, endless trial and error cannot overcome an invariant—a hidden mathematical property preserved or restricted by every legal move.
Conway’s Brilliant Measuring System
John Horton Conway devised and analyzed the puzzle in 1961. Rather than testing every possible formation, he assigned a numerical value to each square on the board. This measuring system is often called a pagoda function.
First, choose a target square in row five and give it a value of 1.
Every other square receives a smaller value based on its distance from the target. The number Conway used was:
[ \sigma=\frac{\sqrt{5}-1}{2}\approx0.618 ]
This is the reciprocal of the familiar golden ratio. Its crucial property is:
[ \sigma+\sigma^2=1 ]
A square one step from the target has weight (\sigma). A square two steps away has weight (\sigma^2), one three steps away has weight (\sigma^3), and so on. Distance is counted by horizontal and vertical steps, not diagonally.
Each soldier contributes the weight of the square it occupies. Add those values together, and you obtain the army’s total mathematical “strength.”
Why the Total Weight Can Never Increase
Imagine a soldier making the most helpful possible move: jumping directly toward the target.
Suppose the landing square has weight (\sigma^n). The jumped soldier and the jumping soldier began one and two steps farther away, so their weights were (\sigma^{n+1}) and (\sigma^{n+2}).
Before the move, their combined weight was:
[ \sigma^{n+1}+\sigma^{n+2} ]
Because (\sigma+\sigma^2=1), this becomes:
[ \sigma^n(\sigma+\sigma^2)=\sigma^n ]
That is exactly the weight of the landing square. The best possible forward jump preserves the total weight but does not increase it.
Jumps sideways or away from the target are no better. They either preserve less useful weight or reduce the total. Consequently, no legal move can make the army’s score rise.
This is the heart of Conway’s proof. The pieces may change position, and the formation may appear stronger, but its measured supply of progress can never grow.
For another puzzle in which simple moves produce deep mathematical structure, explore the Knight’s Tour and its centuries-old chessboard challenge.
The Final Barrier at Row Five
Now for the astonishing part.
If you calculate the combined weight of every square in the complete half-plane below the boundary, measured relative to a target in row five, the total is exactly 1. This follows from adding converging geometric series across the infinitely wide rows, as shown in detailed explanations such as Plus Magazine’s guide to the Solitaire Advance.
However, an actual player must begin with a finite army. A finite formation occupies only some of those starting squares, so its total weight is strictly less than 1.
A soldier standing on the row-five target would have weight 1 by itself. To get there, the army would therefore need to increase its score from less than 1 to at least 1.
But legal jumps cannot increase the score.
The target is unreachable.
This argument covers every finite arrangement, not merely the formations people have already tried. It does not matter whether the army contains 20 soldiers, 2,000 soldiers, or a trillion. As long as the starting force and sequence of moves are finite, row five remains beyond reach.
Why Row Four Is Different
Place the target one row closer, in row four, and the available starting region has more than enough total weight. Conway’s measuring system no longer declares the goal impossible.
That does not automatically provide a solution; it only removes the obstruction. Players must still arrange and move the soldiers correctly. A minimum row-four formation uses 20 pieces, substantially more than the 8 needed for row three.
This difference illustrates an important point about mathematical proofs:
- A construction shows that something can be done.
- An impossibility proof shows that it cannot be done.
- Failure to find a construction proves neither one.
Row four needs a successful construction. Row five needs an argument ruling out every conceivable attempt.
A Small Game With a Powerful Lesson
Conway’s Soldiers belongs to a wonderful family of puzzles with simple rules and enormous consequences. The Tower of Hanoi turns three pegs into a lesson about exponential growth. Conway’s army turns a grid of counters into a lesson about invariants, geometric series, and the golden ratio.
It also demonstrates why proof matters. Thousands of failed attacks on row five might suggest that the goal is impossible, but another formation could always be waiting undiscovered. Conway’s weighting argument eliminates that uncertainty in one sweep.
The soldiers can march sideways, retreat, regroup, and sacrifice one another. The board can stretch farther than anyone could draw, and the starting army can be unimaginably large. Yet every move remains governed by an invisible mathematical budget.
Row four can be conquered. Row five stands untouched—not because nobody has found the right strategy, but because Conway proved that no finite army ever will.
