# Rhombus It

*This game is part of a set of three. We recommend you play this version after having a go at Square It and Parallelogram It.*

*Rhombus It printable sheet**Printable dotted grid*

This game can be played against a friend or against the computer.

Players take it in turns to click on a dot on the grid - the first player will place blue triangles and the second player will place pink squares.

The winner is the first to have chosen four dots that can be joined to form a rhombus.

Rhombuses can be anywhere and any size.

Clicking on the purple settings cog allows you to select the size of the grid, who the players are, and who goes first.

Once you've played a few times against a friend, you might like to discuss your strategies, and then test them by playing against the computer.

**Can you find a winning strategy?***If you are not using the interactive game, you may like to print off some dotty paper.*

You may want to play Square It and Parallelogram It before having a go at this game.

Please be aware that there is a special rhombus which you will know by another name...

Here is a collection of rhombuses, one of which is also a square.

**Therefore a square is a rhombus.**

Well done to Ci Hui Minh Ngoc from Kelvin Grove State College Brisbane in Australia sent in some comments and strategies for this game:

__1 ^{st} Move Advantage:__ Yes, it has the advantage as 1

^{st}player can choose the centre point. For the two games captured shown, when computer is the first player, the computer started with centre point, though this 1

^{st}point did not form the final Rhombus of computer. For the other game, when human is the 1

^{st}player started with centre point, human is the winner and the Rhombus is a square which included the centre point as one of the vertices.

__Strategy:__

I use the properties of Rhombus:

4 sides equal parallelogram. Square is a special Rhombus with angles are 90 degree.

These are additional constraints on parallelogram. That means I must check where the new points are going to place must result “equal length of 4 sides”.

I use the strategy used in creating square, in Square It.

The diagonals of the Rhombus are perpendicular to each other, so strategically, I locate points symmetry around two perpendicular lines (visualise) to maximise the chance to create a Rhombus.

Here are pictures I took after games with computer.

1^{st} player is computer

1^{st} player is human

Why play this game?

This game offers an excellent opportunity to practise visualising shapes and angles on grids and also encourages students to look at strategies using systematic approaches. Describing strategies to others is always a good way to focus and clarify mathematical thought.

### Possible approach

*This game featured in the NRICH Primary and Secondary webinar in April 2023.*

*This printable worksheet may be useful: *Rhombus It

Start with a demonstration playing against a student rather than against the computer. Students may be surprised when the winning rhombus isn't aligned with the grid. This leads to discussions about what makes a rhombus a rhombus.

After a demonstration of the game, students could be left to play for a while in pairs, either on the computer or on a paper grid. Give them the option of reducing the size of the board if they seem overwhelmed!

Bring the class together for a discussion of their thoughts on the game. Did anyone consistently win or lose? Can anyone think of any good strategies which might help them win?

Once ideas have been shared the group can return to playing in pairs, or they can play a game together against the computer, trying, as a class to decide on the best move at each stage. Ask each student to explain the reasoning behind the moves they choose.

One aspect of developing a winning strategy that could be considered is the number of distinctly different starting points ($6$ on a $5 \times 5$ board) and the number of different rhombuses that can be drawn that include each of those points. That is, "Is there a good place to start and why?". This is a great investigation, with the capacity to expand by changing the sizes of the starting
grid, and which leads back into the game itself.

Working on the properties of a rhombus offers an opportunity to look at gradients to establish whether a shape is a rhombus.

With classes who never arrive together or on time, this and other interactive games can be used on the Interactive White Board to engage the early arrivers and set up a relaxed mathematical atmosphere.

### Key questions

- Is your move a good one? Why did you make it?
- Why do you think the computer made that move? Was it a good one?
- How do you know this is a rhombus?

### Possible support

### Possible extension

The computer follows an algorithm (which may or may not be random) to place its pieces.