## 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?

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.

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?

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

or make up their own clues for sets of numbers for others to try.