### Counting Factors

Is there an efficient way to work out how many factors a large number has?

### Summing Consecutive Numbers

Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?

### Helen's Conjecture

Helen made the conjecture that "every multiple of six has more factors than the two numbers either side of it". Is this conjecture true?

# Got It Article

##### Stage: 2 and 3

Published May 2004,February 2011.

Have you tried the Got It game? Got It is an adding game for two. You can play against the computer or with a friend on paper and/or using 23 counters.

This article gives you a few ideas for understanding the game and how you might find a winning strategy. The aim is not to give you an answer but to unpick the problem in such a way that if you meet similar games in the future you might be able to work out winning strategies for them too.

#### The game

The game can be found on the site.

The first player chooses a whole number from 1 to 5.
Players take turns to add a whole number from 1 to 5 to the running total.
The player who hits the target of 23 wins the game.

Here is an example of a game played between two friends, Jo goes first and chooses 5:

 Jo Chris Running Total 5 5 4 9 2 11 5 17 3 20 3 23 Chris wins!

To change the game, you can choose a new Got It target or a new range of numbers to add on.

Are you good at mental arithmetic? Can you play without writing anything down?

#### Working out a Winning Strategy for a target of 23 using 1, 2, 3, 4 and 5

• Try with a target of 23 and a range of numbers 1 - 5.

Is it better to go first or second?
What is the winning strategy?

• If you are not sure, then try a smaller target number, perhaps 13 and fewer "add on" numbers.
Is it better to go first or second?
Can you tell who is going to win before the end of the game?
Does this mean that the game is not about reaching the original target but about reaching another, lower, one?
What is the winning strategy?
Try a smaller target number.  (If you get stuck on this kind of problem, it's often a good idea to choose a simpler example.)
Let the computer go first and look closely at the numbers that it chooses.
Is there any relationship between the number that you choose and the number that it chooses?
Using what you have noticed, have another go against the computer.  This time, you go first.
What is the winning strategy?

#### Try to find a strategy to guarantee winning with any target number using 1, 2, 3, 4, 5.

• Play the game with different target numbers. You may want to try playing games with a target of 9, 14, 37 and 76.

Can one player guarantee winning?  Are the winning strategies similar?
Should you go first or second?

• Now try the same strategy to try to hit the target numbers of 30 or 48.

Does the same strategy work?
Should you go first or second?

Think carefully about the winning strategy that you found when the target number changed, but your choice of numbers was 1, 2 , 3, 4 or 5.

Your strategy probably included the number 6. Why?

#### Can you now find a strategy to guarantee winning with any target number and any range of numbers?

• Can you see what the winning strategy must be for any target number and any range of numbers?

When should you go first and when should you go second?

• If you cannot see what the strategy is yet, then try these games to help you work it out:

We have suggested games with a smaller range of numbers first because starting small often helps to understand the challenge better.
Try games with a choice of 1 or 2.  Play the game with different target numbers.  You may want to try playing with targets of 5, 7, 19 and 21.
Try games with a choice of 1, 2 or 3.  Play the game with different target numbers.  You may want to try playing with targets of 5, 6, 19 and 32.