mathematics

L game

L game is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand L game rather than just read about it. In short: The L game is a simple abstract strategy board game invented by Edward de Bono. It was introduced in his book The Five-Day Course in Thinking (1967).

Key takeaways

L game belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect L game to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of L game from memory before moving on to harder problems.

Reference excerpt

The L game is a simple abstract strategy board game invented by Edward de Bono. It was introduced in his book The Five-Day Course in Thinking (1967).

Description The L game is a two-player game played on a board of 4×4 squares. Each player has a 3×2 L-shaped tetromino, and there are two 1×1 neutral pieces.

Rules On each turn, a player must first move their L piece, and then may optionally move either one of the neutral pieces. The game is won by leaving the opponent unable to move their L piece to a new position. Pieces may not overlap or cover other pieces, or let the pieces off the board. On moving the L piece, it is picked up and then placed in empty squares anywhere on the board. It may be rotated or even flipped over in doing so; the only rule is that it must end in a different position from the position it started—thus covering at least one square it did not previously cover. To move a neutral piece, a player simply picks it up then places it in an empty square anywhere on the board.

Strategy One basic strategy is to use a neutral piece and one's own piece to block a 3×3 square in one corner, and use a neutral piece to prevent the opponent's L piece from swapping to a mirror-image position. Another basic strategy is to move an L piece to block a half of the board, and use the neutral pieces to prevent the opponent's possible alternate positions. These positions can often be achieved once a neutral piece is left in one of the eight killer spaces on the perimeter of the board. The killer spaces are the spaces on the perimeter, but not in a corner. On the next move, one either makes the previously placed killer a part of one's square, or uses it to block a perimeter position, and makes a square or half-board block with one's own L and a moved neutral piece.

Analysis

In a game with two perfect players, neither will ever win or lose. The L game is small enough to be completely solvable. There are 2296 different possible valid ways the pieces can be arranged, not counting a rotation or mirror of an arrangement as a new arrangement, and considering the two neutral pieces to be identical. Any arrangement can be reached during the game, with it being any player's turn. Each player has lost in 15 of these arrangements, if it is that player's turn. The losing arrangements involve the losing player's L piece touching a corner. Each player will also soon lose to a perfect player in an additional 14 arrangements. A player will be able to at least force a draw (by playing forever without losing) from the remaining 2267 positions. Even if neither player plays perfectly, defensive play can continue indefinitely if the players are too cautious to move a neutral piece to the killer positions. If both players are at this level, a sudden-death variant of the rules permits one to move both neutral pieces after moving.

Illustrations

Worked examples

Example 1 — a first encounter with L game

Start with the simplest possible case. Write down what L game claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to L game before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about L game ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of L game

In research: L game appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses L game in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: L game is common in secondary-school and first-year university syllabi. It links to neighbouring topics Abstract strategy games, Board games introduced in 1968, Mathematical games, so understanding it makes those chapters shorter.
In everyday life: Look for L game outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study L game in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what L game means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain L game out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is L game in simple terms?

The L game is a simple abstract strategy board game invented by Edward de Bono. It was introduced in his book The Five-Day Course in Thinking (1967).

Why does L game matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study L game?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on L game.