There is so much potential here for exploration and individual directions whilst practising adding single-digit numbers. The triangle, square, hexagon sequence was deliberately chosen because you have the thirties and forties next to each other and then a little gap to the sixties to force some creative thinking. Whilst there appears to be a lot of choice of numbers when it opens out, actually, very few numbers allow the children to create chains of any length and so this promotes mathematical thinking and careful selection of starting numbers.

**Here is an interactive version of the challenge:**

In this chain, the first number in the triangle includes the digit 3. This is equal to the number of sides/angles.

Choose a number to add to 38 to make a number that could go in the square. It would need to include the digit 4.

+1 | +2 | +3 | +4 | +5 | +6 | +7 | +8 | +9 |

For example:

What could I add to go in the hexagon?

**What if we repeated the shape sequence**? **How many numbers could we have in our chain?**

In this, we are not allowed to add by the same number more than once.

+1 | +2 | +4 | +6 | +7 | +8 | +9 |

In the example above we can go no further. The next integer that has a 3 as a digit is 53. However, we need to add 7 to make that and we have already done that.

Trying other combinations, we can do:

Six numbers in our chain. We can't carry on as we are required to have a number with a 3 as a digit. The only number that is reachable is 63. However, we have already added 1 in a previous step.

**What if we changed the starting number to another two-digit number? **

This opens out the problem quite quickly to a lot of numbers. However, in using their Search skills, they can logically eliminate a lot of numbers. It makes sense to use the tens digit as well as the ones. Therefore, we want to start with a number less than 40.

I have deliberately stated a two-digit number as the problem becomes trivial if we allowed a starting number such as 643,215.

I've found 39 and 23 to be good starting numbers. You can utilise their arguing skills to think about what makes these numbers work better than others. However, a score of nine is possible using another starting number.

There is a lot of scope for exploration in just this. Do starting numbers in the thirties have a similar outcome? What about other tens?

There are number of logical continuations that the children could come up with themselves.

**What if we allowed +0?**

This could potentially be used very strategically to extend a sequence.

**What if we changed the shapes?**

Which set of three shapes allows the greatest number of steps in the chain? What if I used four shapes? This gives them more openness and you can compare whether similar shapes or the different orders of the shapes matter.

A lot of mental addition can be practised from this change alone.

**What if we used subtraction?**

It would be easy for them to assume that with subtraction, the longest chain would be the same length. I'm sure you could get this conjecture from them. Perhaps, the results might surprise them.

**What if we used multiplication?**

I have created a whole new set of isles for this possibility. Check them out here.

**What if we had ten-sided shapes?**

Then we would be required to have the digits 1 and 0 in that order within the number. I'm not really sure how that would work in terms of a chain of any length but I have put it in here as the kind of thing a child may come up with. It is still worth wondering about even if it is just for a minute or two before realising that the pathway does not lead very far.

**Children's possible use of reasoning skills:**

## Search

It is important for them to realise why Roaming will not be very successful in choosing a starting number. This is an island that promotes the use of Seeking. They have to make use of the thirties, forties and sixties but the additional challenge of only being able to use each addition once means that only a few numbers work very well.

This also then extends to if they use different shapes. Why would some combinations not work as well?

## Organise

If you wanted to, you could get them to investigate lots of two-digit numbers and have a high-score chart on a hundred square. This might help them to Seek a good starting number if they are struggling to do this.

## Discover

Conjectures are possible on the length of chains possible. Will making changes likely increase or decrease the length of a chain? Each time they change the rules, they can think about what impact this will have. If the goal is the longest chain, this can make them consider how they go about choosing new pathways.

## Investigate

If the goal is the longest chain, by making changes to starting numbers/shapes, they are then investigating their belief that it will result in a longer chain. As such there is a lot of scope for investigation once they get beyond the initial conditions.

## Argue

Lots of potential here for them to try and explain why some starting numbers/shapes will not achieve longer numbers. Some of that will be internalised in their strategising but as teachers, we can open it out to discussion.

They can also think about why they cannot continue a chain or why it is hard to go beyond a certain length.

## Explore

The children can definitely suggest changes to the rules to take the inquiry further. There are lots of possibilities here.

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