Dots and Boxes Strategy: When to Leave Two Boxes

Follow the extra turn

This Dots and Boxes strategy uses ProjectGAMR's two-player board of four rows and four columns of dots. It contains nine boxes. A line that completes at least one box scores for the person drawing it. That player moves again unless the board is complete. A line that scores zero passes the turn.

A chain connects unclaimed boxes through missing shared edges. Once someone opens an end, a scoring move can expose the next box and supply the extra turn needed to claim it. Taking the entire chain can leave its collector responsible for opening the next row. The following position makes that decision visible.

Replay the position on the browser board

Count dot rows downward from the top and columns rightward from the left. Use the left endpoint for a horizontal line and the upper endpoint for a vertical line. H13 goes right from the third dot in the top row. V24 goes down from the rightmost dot of the second row. Start a new game with Lime drawing first.

Draw H13, H21, H22, H23, H31, H32, H33, H41 and H42 in order. Then draw V11, V12, V13 and V14. Lime claims the top right box. Draw H43, H11 and H12. Violet claims the other two top boxes and keeps the turn.

Violet draws V21 to open the middle row from the left. Lime draws V22 to finish the first middle box. The score is now 2–2, and Lime still has the turn. Use this checkpoint for both continuations.

Collecting the middle boxes opens the last row

Lime can draw V23 and then V24, claiming the two middle boxes still available. Lime leads 4–2 and must draw again. The only unused lines are in the bottom row, where each box already has its upper and lower sides.

Try V31 as Lime's next line. Violet can take the bottom boxes from left to right with V32, V33 and V34. Violet wins 5–4. Those two immediate middle boxes came with another move for Lime, which gave Violet the last chain.

Close the far side to hand over a pair

Start a new game and replay the setup through V22. As Lime, draw V24 before V23. This far edge scores zero because both middle boxes are still missing their shared side. That shared side, V23, can now complete both with one stroke.

The zero-score move passes the turn to Violet. In his 2021 analysis, mathematician Daniel Allcock describes this idea as the hard-hearted handout. Leaving a pair in the open chain keeps the next opening with the opponent if they take the pair. Here the shared side and all four vertical edges in the bottom row remain available.

Check both of Violet’s replies

If Violet draws V23, Violet collects both middle boxes and leads 4–2. The extra turn requires a bottom-row move. For example, Violet can draw V31. Lime then takes all three bottom boxes with V32, V33 and V34, winning 5–4.

Violet could also open the bottom row before taking the pair. That leaves Lime free to collect both middle boxes and all three bottom boxes. For a replay, let Violet draw V31 after the handout. Lime can draw V23, V32, V33 and V34 to win 7–2.

Look for the move the pair buys

Before handing over boxes, check that the proposed edge scores zero and that the pair can be taken with one line. Then inspect the remaining board. Here Violet must open the bottom row after accepting because it contains every unused edge.

If you are already collecting the final chain, finish it. There is no later opening to pass back. In another position with more unclaimed regions, inspect what the opponent could open before trading away boxes. Use New game to replay both branches on this board and compare who draws the first bottom-row edge.