Infinite Connect Four

Shall we play infinite Connect Four on the expansive infinite board? What size winning chains might we aspire to make? What are the winning strategies? You may be surprised.

Consider the familiar game of Connect Four, shown here. Starting with an empty 7 × 6 grid, players take turns dropping colored coins into it. When a coin is placed in a column, it falls under gravity to occupy the lowest available cell. The first player to achieve four of their coins adjacent in a row, a column, or on a diagonal wins the game.

In the sample game in progress here, note how several four-in-a-row attempts have already been blocked during play. It is currently Yellow's turn to play—where should she play? In fact, she can win the game in three moves. Can you find it?

Connect Four is known to be a first-player win with optimal play on the standard 7 × 6 board—the first player can win on their 21st move (that is, with 41 moves in all, leaving only one cell unplayed), when opening with a move in the center column. Opening in the center-adjacent columns (3 and 5) leads to a drawn outcome with perfect play, while the other four opening moves lead to a loss.

Connect infinity

I shall naturally be seeking, however, to explore the infinitary variations of Connect Four. The game works perfectly well when played on the upper half-plane, for example, a playing board arranged like the integer grid ℤ × ℕ, with a column for every integer, each column a copy of the natural numbers going up. The players take turns dropping their coins into the columns, which fall under gravity to occupy the lowest available cell.

But what should the winning condition be? Shall we still take four-in-a-row as sufficing to win? In light of the luxurious possibilities of the infinite board, that would seem downright miserly—there is plenty of room, after all, for far longer winning chains. The players will naturally aspire to form very long winning chains. Is it possible? What length chains can they form? Can the players play so as to form arbitrarily long finite chains on a row, column, or diagonal? Can they reasonably aspire to make infinite} winning chains?

In this week’s installment, we look in the various infinitary variations of Connect Four, now played on an infinite board. Finite Connect Four on the 7 × 6 board is a first-player win, but what about infinite Connect Four? What size winning chains might we aspire to form on the expansive board? Infinite Connect Twenty? Infinite Connect Infinity? You may be surprised at the final answer. We shall also see an application of infinite Connect Four to finite Connect four with passing.

The post includes seventeen custom tikz images for illustration.