Which Numbers? (2)

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Which Numbers? (2)

This problem is similar to Which Numbers? (1), but slightly more tricky. You may like to try that one first.

I am thinking of three sets of numbers less than $101$. They are the blue set, the red set and the black set.
Can you find all the numbers in each set from these clues?

These numbers are some of the blue set: $26, 39, 65, 91$, but there are others too.
These numbers are some of the red set: $12, 18, 30, 42, 66, 78, 84$, but there are others too.
These numbers are some of the black set: $14, 17, 33, 38, 51, 57, 74, 79, 94, 99$, but there are others too.

There are sixteen numbers altogether in the red set, seven numbers in the blue set and fifty numbers in the black set.

These numbers are some that are in just one of the sets: $6,10, 15, 24, 33, 48, 56, 65, 75, 93$, but there are others too.
These numbers are some that are in two of the sets: $12, 13, 36, 54, 72, 96$, but there are others too.
The only number in all three of the sets is $78$.
These numbers are some that are not in any of the sets: $5, 8, 22, 27, 44, 49, 63, 68, 82, 86, 100$, but there are others too.
There are twelve numbers that are in two of the sets.
There are $41$ numbers that are not in any of the sets.

You can download a sheet of all this information that can be cut up into cards.

Can you find the rest of the numbers in the three sets?
Can you give a name to the sets you have found?

Why do this problem?

This problem requires learners to see the connections between numbers in a set and so find the rest of the set. They will need to make and test hypotheses, and justify their reasoning.

Possible approach

You could start by using this interactivity during a 'warm-up' activity. Choose a particular property and drag one number with that property onto the left side of the grid. Invite the group to work out what the property is that you have chosen by calling out other numbers, which you then place on the appropriate side of the grid. Can they decide upon the 'rule' in as few guesses as possible?

The class could then work in pairs on the problem itself so they can talk through their ideas with a partner. They will need this sheet of information about the numbers, which can be cut up into cards to make it easy to use. Observing how children record as they work on this challenge will be very informative for you. This special hundred square could help some learners to record if they are struggling to find their own way. The different groups of numbers, the red set, the blue set and the black set could be recorded like this:

Using a hundred square to record (whether it is the special one or a 'standard' one) will reveal patterns and therefore may help children work out their properties.

Discussion at the end of the lesson could include not only the sets of numbers that have been found, but also the ways that the children approached the problem. What did they do first? What were their first ideas? How did they decide whether these initial hypotheses could be right? How did they record their thinking? Did they work in a systematic way? How did they know that their solution was correct?

Key questions

What is the same about these numbers? What do these numbers have in common?

What do you know about this number? Is that true of any of the others in the set?

How will you keep track of your thinking?

Can you see a pattern on the hundred square?

Can you see any gaps in that pattern?

Why do those numbers make that pattern?

Possible extension

Learners could do the much harder Ben's Game or make up their own clues for sets of numbers for others to try.

Possible support

Some learners might need you to suggest using a hundred square and they could start by putting the numbers in the "red set" onto it. Can they find the rest of the numbers in this set? They might start by trying this slightly more straightforward related problem.