This is related to the "guess my number" activity, but comes from a different angle, and has a surprising connection to binary numbers. The chart below will be used from activity steps 9 onwards. You can print this out, copy it off the computer or just show it on a screen

- At least 2 people
- A pen or pencil
- 8 small pieces of paper (you can cut a sheet into 8 parts)

- Comparing numbers from 0 to 7

If you have a pile of 8 cards, and you remove half, then half again, how many times can you do this until you get to just one card?

Step 1

Let's cut out 8 cards, ...

Step 2

... write the numbers from 0 to 7 on them, and put them out in order.

Step 3

I’m thinking of one of the numbers. You can give me a number, and I’ll tell you if your guess is greater than or equal to the one I’m thinking of.

In this case the presenter is thinking of the number 6; the child has chosen card 3, so 6 is greater than or equal to 3.

Step 4

(Depending on your number)

My number is greater than or equal to your guess, so which cards can you eliminate?Or

My number is not greater than or equal to your guess, so which cards can you eliminate?

If your secret number is greater than or equal to their choice (as in the photo), the child can move all smaller cards to one side, to show how it is narrowed down (ideally the child can work out which cards can be eliminated; they need to be careful to be accurate!) If their guess is not greater than or equal (i.e. is smaller), they can eliminate the card with their guess and all larger cards.

Step 5

What’s your next guess?

Continue this until one card is left; sometimes you will eliminate all cards to the left of their guess, and other times it will eliminate all cards to the right including their guess (that’s because of the "equal" in "greater than or equal"). In this case, the child has chosen card 6 (eliminating everything to the left of 6)...

Step 6

What’s your next guess?

and then 7 (which eliminates the 7, leaving only 6 behind as the correct answer).

Step 7

You can try this for other numbers while they refine their strategy. Guide them towards the strategy of choosing a number that will divide the list into exact halves (they should choose 4 first, then either 2 or 6, then either 1, 3, 5, or 7). It’s best if they gradually work this out from a few games, rather than being told the rule.

Step 8

When there are just two cards, asking if it is "greater than or equal to 1" is the same as asking if it is 1, but it’s good to enforce that the only question allowed is "greater than or equal to".

Step 9

Here’s a chart that will help you to ask questions. I’m thinking of a new number from 0 to 7. Start at the top, and follow the "yes" or "no" line as I answer each question.

The child might recognise this as being the same strategy they have learned just before. Guide them to ask if your number is greater than or equal to 4 first, and then follow either the "yes" or "no" branch down the chart.

Step 10

In this case the presenter is thinking of the number 1, so the first answer was "no", and now the child asks if the number is greater than or equal to 2.

Step 11

Continue until they reach the bottom of the chart. That should tell them the number you’re thinking of. In this case, the sequence of three answers was "no, no, yes" to get to the answer, which is 1.

Step 12

We can do this without you having to ask the questions. If I give you the answers "yes, no, yes", where do you end up?

Help the child to see the path to 5.

Step 13

How about "no, no, no""

They should get to 0.

Step 14

Yes, yes, yes?

That gets to 7.

Step 15

We’ve figured out a way to communicate numbers just by saying yes and no!

If you’ve done the binary challenge activity, they may realise this is a different way to achieve the same outcome - the three yes/no choices are the same as saying whether or not to flip over each of three cards with 4, 2, and 1 dots on them respectively. The chart that we used is called a "decision tree".

This way of thinking about answering questions has led to a way of understanding how all data is represented on computers using the binary number system. If you do the "binary challenge" activity, you’ll experience the same thinking in a different way. The way we have done it here relates to the area of computer science called "information theory", where we think about how to represent data using just two different values. When we represent data, we often represent "yes" and "no" as the digits 1 and 0; in fact, the use of these two digits is the reason we call our devices "digital". The idea that you can communicate numbers and words by just saying yes and no is closely related to the idea that we can represent any data using combinations of just two different digits.

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Arnold's Challenges