When we dig down… excavating the root, we notice that each new number encountered is the sum of two numbers that are on the root-path starting at 1 and dropping down.

For example, 5 = 2 + 3 and 6 = 2 + 4

This looks like it should work because 6 = 3 + 3 (also because 6 = 5 + 1), but alas the tree has not revealed all the rules governing root growth…

The second rule of root growth is that each number must be as high as possible. This 6 is too far down in the earth. It can be one level higher.

It can be at this level.

6 = 3 + 3

After a strenuous day of work we have exposed a lot more of the giant Banyan root. Unfortunately, when writing this down in our note book, we’ve made a mistake. Find the number that can be at a higher level.

Of course 12 = 5 + 7, so the addition is correct, but the 12 is too low.

We could put the 12 here. That’s one level higher and is actually as high as the 12 can get.

Is this correct?

No, this time the 11 is not in a good place.

Of course 11 = 8 + 3

…but the 3 is on the other side of the root so it cannot be used in a sum to make 11. Only two numbers that are part of the root-path dropping down from “1” may be used.  On the rightmost root-path we have {1,2,4,8} no two of which add to 11.

Strong and healthy Banyan roots will have the number 15 at level 6. That’s the best possible. However, some Banyan roots are not that good. The Banyan tree pictured here is sick. The 15 is at level 7. It cannot be placed higher without rearranging the roots…

Level 1: {1}   Level 2: {2}   Level 3: {3, 4}   Level 4: {5, 6, 8}   Level 5: {7, 9, 10, 12}   Level 6: {14, 13, 11}

This is a root from a stronger Banyan tree than on the previous slide. From 1-14 both are equal, but at 15 this one is superior by getting the 15 up higher.

Level 1: {1}   Level 2: {2}   Level 3: {3, 4}   Level 4: {5, 6, 8}   Level 5: {7, 9, 10, 12, 16}   Level 6: {14, 13, 15, 11}

The way I presented the tree roots is not perfect. The levels are not as obvious as they should be. The puzzle-sheets for students have the levels more obvious.

You may want them to use different colors for the different levels as this slide.

(You will be able to download the puzzle sheets on the last slide.)

This shows the highest level possible for each number 1-50. Only the number 47 needs nine levels.

Digging down with numbers 1-50 find the best possible Banyan tree.

Spoiler: On the next page you will find a tree that has all numbers as high as possible. Do not show the next page to your students or even hint at its existence 😉

If you are googling this puzzle look for  “minimal addition chains.”

This is an unbelievable Banyan tree. All numbers are as high as they possibly can be.

I wonder – is it possible that one such root system could hold all the integers? I don’t know.

Do not show this to your students until they have thought about it for a week.

Banyan tree roots can be made competitive and/or co-operative.

Competition may be between desks, but it can also be between classes. I sometimes do puzzles simultaneously with two classes – that’s a nice blend between co-operation and competition.

Co-operation within a class can have students with different roles. One of my favourite roles is to have an error-finder. This is an average student who goes around trouble shooting other student’s solutions (when requested).

There are two puzzle-sheets. You can download them both here.