Zebra

Herds multiply together, so three threes beat one nine — and a stone left on its own throws itself away.

Grades: 3–6
Subjects: Multiplying & Dividing, Logic, Shapes & Space
Kind: Two-player game
Play: Against the computer, two on one tablet, or two online
Boards: 21: sixteen against the computer on seven fields, five for two players
Length: 5 to 20 turns, depending on the field
On paper: Printing a board from the browser gives a blank field and the rules for two pencils
Cost: Free, no account, no ads
Original: Zebra at MathPickle

What it is

A field of hexagons, striped black and white like the animal. You lay two stones a turn, and either colour may be the one you lay. When the last cell fills, your herds — stones of your colour that touch — are multiplied together, and that product is your score.

The rules

A zebra's pelt is striped black and white. When the pelt is complete, Black scores by multiplying together the sizes of all its groups. White does the same. The highest score wins.

The first player is Black. They play a single stone. The second player is white. They play two stones. Turns alternate. Two stones are played each turn after black's first turn.

When playing a stone, you may choose to play white or black.

Why would you want to play your opponent’s colour? Because joining two of their groups can shrink their score.

Zebra was made by graduate students at Dalhousie University, after Nick Bentley's game Produto.

What it gives you to practise

Multiplication as a reason rather than a result. Three herds of three score 27 and one herd of nine scores 9, so bigger is worse, and a stone left on its own multiplies everything by one and throws the turn away. The best thing on the board is why you would ever lay the rival’s colour: fill the gap between two of their herds of three and the two fuse into a single animal of seven, so 3 × 3 = 9 falls to 7. Sometimes the deadliest stone you lay is theirs.

Twelve stones

Say one colour ends the game with twelve stones on the field. Its score depends only on how those twelve are split into herds.

Herds	Multiplied	Score
12	12	12
6 + 6	6 × 6	36
3 + 3 + 3 + 2 + 1	3 × 3 × 3 × 2 × 1	54
4 + 4 + 4	4 × 4 × 4	64
2 + 2 + 2 + 2 + 2 + 2	2 × 2 × 2 × 2 × 2 × 2	64
3 + 3 + 2 + 2 + 2	3 × 3 × 2 × 2 × 2	72
3 + 3 + 3 + 3	3 × 3 × 3 × 3	81

Threes come out on top. A herd of four is worth exactly what two herds of two are worth. A herd of five or more is always worth less than the same stones cut into twos and threes: five scores 5, but two and three score 6. A lone stone multiplies by one, so it adds nothing and still uses up a cell.

On the field, threes are not free. Two of your herds stay apart only if the cells between them end up the other colour, and your rival lays stones in those cells too.

In a classroom

Against the computer: Four rivals, weakest first, each named for what it does: Plays at Random, Builds Two Big Herds, Counts Only Threes, and Many Herds Against One. Every rival was played against every other on the same fields, and each loses to the one after it. The three stronger rivals have no chance in them, so a lost game can be played again move for move until the turn where it went wrong turns up.
Two on one tablet: Five fields for two players sharing a screen, kept because they are fair. The computer played each one 48 times against itself, and Black’s share of the points stayed between 35% and 65%. Games run 9 to 16 turns.
Online: One player makes a room and reads out its code or sends its link; the other joins on their own device. They play the same five fields.
On paper: Print a board from the browser and the sheet says how to play it with two pencils: one player shades cells, the other outlines them.

One way to spend a lesson on it

Show the four rules pictures on a projector and read the sentence under each one.
Pairs share a device and play Plays at Random on The Path, then on Three by Three. Each game is five turns.
Stop the class and ask the twelve-stones question: which scores more, one herd of twelve or four herds of three?
Pairs play each other on Grazing Ground, which takes nine turns, then compare their herds and their scores.
Whoever beats Plays at Random on all four of its fields moves on to Builds Two Big Herds.

Questions to ask

Before the first game

Which is worth more, one herd of six or two herds of three?
Two threes: 3 × 3 = 9, against 6. Most of the game follows from this.

While they play

Could your rival join two of your herds in one turn?
A turn is two stones, so a gap of two empty cells is no safer than a gap of one.
Is there a cell where a stone of your rival’s colour would hurt them?
Look for an empty cell that touches two of their herds. Filling it with their colour turns 3 × 3 = 9 into one herd of 7.
Who has laid more stones so far?
Black opened with one and every turn since has been two, so after each turn the player who just moved is exactly one stone ahead. Neither player is ever more than one stone up.

After a game

With the same stones, how would you cut them into herds now?
Counting the stones and trying cuts on paper is the twelve-stones table again, with their own numbers.
What did the rival do to beat you?
Each rival is named for its plan. Many Herds Against One puts its stones along the edge, where a cell has fewer neighbours, so its herds are harder to join.

Where people get stuck

Building one big herd: The herd gets bigger, so it feels like winning: adding, where the game multiplies. Ask for the score out loud, then ask what the same stones would score as two herds.
Never laying the rival’s colour: Tapping a cell twice lays the other colour. Ask where a stone of the rival’s colour would do them harm.
A one-cell gap between two of their own herds: The rival lays a stone of the player’s colour there, and the two herds become one. Ask them to look at the field from the rival’s side before each turn.
Lone stones: A lone stone multiplies the score by one. It adds nothing and uses up a cell that could have grown a herd.
Losing count at the end: Products grow fast: 3 × 3 × 3 × 3 is already 81. Pairing helps, since 9 × 9 is easier than working left to right.

Going further

Why threes? Ask for the best cut of ten stones, then of eleven. They are 3 × 3 × 2 × 2 = 36 and 3 × 3 × 3 × 2 = 54. The pattern holds for any number: use as many threes as possible, finish with a two or two twos, and never leave a one.
Is the opening fair? It depends on the field. With the computer playing both sides, Black wins every game on The Path, a straight row of nine cells, and White wins three games in four on The Anthill. The fields that came out even are all rhombi and triangles, shapes with no narrow waist and no ring. Ask why a narrow waist might hand the game to one side.
Balanced Zebra is the same field with no opponent. Paint every hexagon so both colours multiply out to the same number, then to the largest the shape allows.

Curriculum

CCSS 3.OA.C.7: Fluently multiply within 100. Every score is a product, worked out at the end of each game.
CCSS 3.OA.B.5: Use properties of operations as strategies: 3 × 3 × 3 × 3 regrouped as 9 × 9.
CCSS MP7: Look for and make use of structure: why threes beat every other cut.

Play Zebra