Have you got it?
Can you explain the strategy for winning this game with any target?
Problem
Have You Got It? printable sheet
Have You Got It? is an adding game for two players. You can play against the computer or with a friend. It is a version of a well known game called Nim.
Start with a target of 23.
The first player chooses a whole number from 1 to 4.
Players take turns to add a whole number from 1 to 4 to the running total.
The player who hits the target of 23 wins the game.
Play the game several times.
Can you find a winning strategy?
Can you always win?
Does your strategy depend on whether or not you go first?
To change the game, choose a new target or a new range of numbers to add.
Test out the strategy you found earlier. Does it need adapting?
Can you work out a winning strategy for any target?
Can you work out a winning strategy for any range of numbers?
Is it best to start the game? Always?
Away from the computer, challenge your friends:
One of you names the target and range and lets the other player start.
Extension:
Can you play without writing anything down?
Consider playing the game where a player CANNOT add the same number as that used previously by their opponent.
You may want to take a look at Last Biscuit, a more demanding game which requires similar strategic thinking.
Click here for a poster of this problem.
Getting Started
Work backwards.
Simplify the game by choosing a smaller target and working out a winning strategy.
Play the game against the computer and think about the computer's strategy.
Student Solutions
Adhvik from Inventure Academy in India (buried under other stuff), Daniel from Wordsley in the UK, Miss Kelly's 7a3 students from Acklam Grange School in the UK, Aayan from Robert Gordon's College in the UK all noticed a number that helps you win. Aayan wrote:
First player to get 18 has the chance to win.
Daniel said this in a different way:
In order to win you would have to get the person to get to 19
Shaunak from Ganit Manthan, VicharVatika in India explained why:
I played several times with the computer and observed that it was defeating me when it achieved $18.$ I realised this would always work because $23 - 5 = 18,$ and if I play any number from $1$ to $4$ called $x$, the computer subtracts $x$ from $5.$
Dylan from Forest School in the UK, Ci Hui from Queensland Academy of Science Mathematics and Technology (with diagram), Miss Kelly's 7a3 students and Shaunak extended this to a more complete strategy. Miss Kelly and her students wrote:
You always need to get the number 18 before your opponent.
Look at opponents last score.
Work backwards.
If both opponents know to get 18, if they get to 13 before you then they
have won, as no matter what numbers you use, they can still make 18.
For example, 3, 4 (7), 3 (10), 3 (13)..... 1(14), 4 (18), 1 (19), 4 (23).
Dylan wrote:
I realised that I needed to get the AI to play on 18, 13, and 8 so I realised that if I started at 3 then then AI would to play something to get me to 8 and then I coukd win from there.
Ci Hui used a diagram to explain the idea (click on the image to open a larger version):
Adhvik, Ci Hui, Miss Kelly's 13 year old students from Acklam Grange School and Shaunak explained how their strategies could be adapted for other sets of numbers. Click here to see Shaunak's full video solution on YouTube.
This is Adhvik's work (click on the image to open a larger version):
Miss Kelly and her 13 year old students wrote something similar, but their strategy starts from the very first turn:
We can always win, if you start first or if your opponent does not know the trick.
If we add the biggest and smallest number (from the list of numbers in which we can add), we subtract this from the target and continue subtracting until we get to the smallest number.
For example, if we can add the numbers 1, 2, 3, 4, 5 and the target is 29. The first step is 1 + 5 = 6, we then subtract 6 from 29 to get 23, and continue until we get to the lowest we possibly can...... 29, 23, 17, 11, 5. We need to get to all of these numbers (which are our milestones), for us to definitely win.
This works for any set of numbers and targets.
Ci Hui showed some more examples and used another diamgram to explain why this method works for finding winning strategies (click on the image to open a larger version):
Teachers' Resources
Why play this game?
Have You Got It? offers an engaging context which requires students to use simple addition and subtraction to find a winning strategy. The challenge involves working systematically and strategically, conjecturing, refining ideas, generalising, and using knowledge of factors and multiples.
Possible approach
Key questions
How can I work out the 'stepping stones' that I must 'hit' on my way to the target?
Possible support
You can alter the settings on the game to have a lower target and a shorter range of numbers (for example a target of $10$ using the numbers $1$ and $2$). As you play, note down the running totals to refer back to later.
Possible extension