Latin Lilies
Problem
This is a game for two players, which can be played using the interactivity below. The default setting allows you to play against Computer 1 ('Capable Computer'), but if you click on the purple cog, you can change the settings and play against a friend or against Computer 2 ('Crafty Computer') or Computer 3 ('Champion Computer').
Rules of the game:
- Each player is trying to get frogs to hop up or down a column from their side of the pond to the other side.
- This is possible if a frog can hop onto three numbers that increase in size in their direction of travel. Frogs are allowed to hop over other numbers and spaces on the way.
- Players take turns to place an available number (the lily pads) into any space on the grid (the pond). Numbers can only appear once in each row and can only appear once in each column.
- The winner of a column is the first player whose frog is able to hop across the pond, even if the final number has been placed by their opponent. So watch out that you don't give your opponent a column by mistake!
- When a player wins a column, it will be marked with a star and cannot be won by their opponent.
- The aim of the game is to win more columns than your opponent.
You will notice that you start with frog spawn on your side of the pond and as you place numbers that could be used to form your path to the other side, you'll see different stages of the frog life cycle illustrate how close you are to winning each column.
Depending on the previous moves made, it may not be possible to place all of the lily pads into the pond.
Can you develop a strategy that ensures you regularly beat 'Champion Computer' (Computer 3 in the settings menu)?
Further challenges:
Tea Cups, Nine Colours and Latin Numbers all offer different contexts in which to apply similar mathematical thinking.
This problem was inspired by Jorge Nuno Silva's presentation at the Big Internet Math-Off 2019. We are grateful to Jorge, who kindly gave NRICH his permission to publish our version of his game.
Teachers' Resources
Why play this game?
Latin Lilies can be used as an introduction to Latin squares. It offers an engaging context in which students are challenged to plan ahead and devise winning strategies.
Possible approach
Introduce the game using the interactivity with you playing against Computer 1 (the default setting). You could choose to play in silence and invite students to watch the game, and consider what the rules might be. Give them a chance to talk to a partner and then draw everyone together to share suggestions and clarify how to play.
Once the rules have been established, invite students to play in pairs against Computer 1 on a tablet or computer. Once they are confident they can beat Computer 1, they can be challenged to take on Computer 2 and then Computer 3.
If students do not have access to a tablet or computer, they could play the game against each other using paper and pencil.
In the plenary, you could ask students to articulate their strategies, perhaps in the context of offering advice to a friend about how to play the game, and/or invite students to test out their strategies by demonstrating they can beat Computer 3 or by playing against the whole class.
Key questions
Which of the five numbers is it best to place at the start?
How many columns are you going to try and win?
Can you ever block your opponent so that they cannot win that column?
Possible support
The task Three by Three will help students focus on the properties of Latin squares.
Possible extension
Tea Cups, Nine Colours and Latin Numbers could all be offered as challenging follow-up tasks.