Start from the end
Problem
There are 21 stones. Two players take turns removing 1, 2 or 3 stones. Whoever takes the last stone wins. You move first. What should you take?
Take 3 — the most allowed — to get closer to the end.
Take 1, leaving 20. Then always leave a multiple of 4 and you cannot lose.
Show the worked solution
Step 1 — the end position. If 0 stones remain on your turn, you have already lost: your opponent took the last stone.
Step 2 — positions 1, 2, 3. You take everything and win. These are winning positions for the player to move.
Step 3 — position 4. Whatever you take (1, 2 or 3), you leave 3, 2 or 1 — all winning for your opponent. So 4 is a losing position.
Step 4 — the pattern, with its reason. From any position, you can reach the next multiple of 4 below it unless you are already on one. Whatever your opponent removes from a multiple of 4 (call it k), you remove 4 − k, restoring a multiple of 4. Losing positions for the player to move: 0, 4, 8, 12, 16, 20.
Step 5 — answer. From 21, take 1. Test the limit: if the pile started at 20, the first player would lose against perfect play.
This method is called backward induction. It replaces "what feels aggressive?" with "which positions are safe to hand over?" The same idea underlies Nim, analysed by Charles Bouton in 1901, and the Sprague–Grundy theory of impartial games.
Go: influence, liberties and the limits of counting
Go — weiqi in China, baduk in Korea, igo in Japan — has documented roots in ancient China and spread to Korea and Japan in the first millennium CE. Players place stones on the 361 intersections of a 19×19 grid. A group is captured when it has no liberties: no empty adjacent points.
A tempting assumption for beginners is that capturing stones is the goal. In fact the score counts territory (and, under some rules, stones on the board). A stone that "lives" quietly can be worth more than a capture that leaves your own group thin. Strategic logic in Go is about trading local gains for global shape.
The ko rule shows how a game designer handles infinite loops: a player may not immediately recapture in a way that recreates the previous board position. Without it, a single exchange could repeat forever.
In March 2016, DeepMind's AlphaGo defeated professional Lee Sedol 4–1 in Seoul. Its 37th move in game two was widely described as one a human professional would rarely consider — a reminder that strong strategy sometimes looks wrong at first glance.
Shogi and xiangqi: same family, different consequences
Both descend from the broad chess family whose early form is linked to chaturanga in India. Their rules create very different strategic logic.
| Feature | Shogi (Japan) | Xiangqi (China) |
|---|---|---|
| Board | 9×9 squares | 9×10 points with a central "river" |
| Captured pieces | Can be dropped back into play by the captor | Removed permanently |
| King / general | Moves freely one step | Confined to a 3×3 palace |
| Strategic effect | Material rarely "simplifies"; attacks stay sharp to the end | Exchanges reduce the game toward technical endgames |
Test an assumption from Western chess: "trading pieces when ahead is safe." In xiangqi that often holds. In shogi it can backfire, because every piece you capture can be dropped by you — but every piece you give up can be dropped against you. Consequences depend on the rule set, not on a universal maxim.
Sowing games: congkak and pallanguzhi
Mancala-family "sowing" games are played across Asia and Africa. In Malaysia, Indonesia and Brunei, congkak (also congklak) uses a carved wooden board with rows of holes; in Tamil Nadu and Kerala, pallanguzhi uses fourteen pits. Players pick up all seeds from a hole and sow them one by one into following holes.
The strategic lesson is exact counting ahead. A hole holding seven seeds is not "more powerful" than one holding three; what matters is where the last seed lands, since many rule sets let you continue sowing or capture based on that final hole. Strong players count landing positions rather than seeds.
Source context
Rules vary by region and even by village; descriptions here follow commonly published versions. For the history of board games in these regions, see the puzzle traditions page.
Ready to add uncertainty to your decisions? Continue to probability puzzles.