Worms
Problem
Imagine a square patch of ground in a garden, and because it's a good garden there are worms around, and they feed from the soil. Well, I've taken an imaginary square and given it the numbers 1 to 100.
You've probably come across number squares like this in maths sometimes at school. So each part of this square piece of ground has a number. But now, suppose that the centre piece of this square is what the worm likes to eat most of all. So I've taken the centre 16 squares and made them a little darker.
You will probably notice that the lowest number in this square is 34 and the highest is 67. Well, that's enough about the ground. What about the worm? The worm will be 4 squares long and to make sure we know which is the head and which is the tail I've coloured the head part differently.
Of course the worm can bend itself in different ways. But I'm going to suggest that on this occasion it has 4 square parts [like 4 multilink] which have to be joined edge to edge in the "normal'' way. Here are three examples that would be O.K., and one which is not because the head is not at the end of the body!
Now, to set up the challenge. With a square in front of you, with the numbers 1 to 100 marked on as shown and the centre 16 squares marked in some special way, place your worm, perhaps made out of multilink, on the square. [Multilink are 2 cm long so get hold of some 2 cm squared paper; the backing paper for some "taky-back'' film is just the right size.] When you place your worm you have to make sure that the head is on one of the centre squares that are shaded differently. The head stays there, but while the worm is eating it wriggles and changes the shape of its body, in as many different ways as it can. Here are just three of the many ways in which the worm can be, and I've placed its head on the number 34.
The challenge is to:-
- Find all the shapes and positions that the worm can be with its head fixed in the one place [in my example, on number 34, but you can choose any of those 16 squares]. Don't forget about the worm's body going North, South, East and West.
- Add up the numbers that the worm is on, that is the numbers that the head and the rest of the body are resting on. [In my first example 34 + 33 + 43 + 53 = 163; 2nd example gives 34 + 33 + 23 + 13 = 103; 3rd example gives 34 + 33 + 23 + 22 =112.]
- Find a good way of recording your results
- so that you can make sure you do not do the same one twice and
- so that you can compare the results easily.
Well, that's really it! Of course at the end you have to say "I wonder what would happen if I ...?''.
Getting Started
You could try keeping the second and third body parts in one place to begin with. How many different places are there for the fourth part?
What happens if you still keep the second part in the same place and investigate different arrangements of the third and fourth now?
You might find this 100 square useful.
Student Solutions
Thank you to everyone who sent in their solutions for Worms.
Poppy from Dussindale Primary School sent in this excellent solution and took it further with her investigation into number patterns:
I started with the head of the worm on the number 34.
First of all I began by trying out the different shapes. I rotated each shape through north, east, south and west. I then realised that some of the shapes had a reflection. I started with the L shape and noticed whilst some of the numbers were different that there were repeated totals. I began to investigate why this happened - see the algebra for each of the L
shaped totals. I then looked at the S shaped worm and saw that the reflections came up with repeated totals. I found that there 16 different totals but 24 different calculations that could be made by rotating or reflecting the worms.
From Springvale School we had a number of solutions sent in.
Hannah, working with snakes instead of worms!
I worked out that to get the smallest number for each one, you have to point the snake directly up, and if you want to get the highest possible number for each one you needed to point the snake directly down. I also worked out that there are 32 different positions for each square, that's 512 in total! I know this, because there are different ways the worm can
be positioned and you can put each one, North, East, South, West.
Meg
I worked out to get the smallest possible solution, you need to put the worms head on 34 (the smallest number on the shaded grid) and its body above it so it sits on 34,24,14 and 4 which equals = 76. To get the biggest solution you need to put the worms head on 67 and the body below it on 67,77,87 and 97 this = 328. I worked out that there are 32 solutions for each box. I
found this out because there are 8 different ways that the worm can be positioned and you can make 4 worms, so 4x8=32.
Martha
I worked out that there are 512 solutions for worms....
I know this because there are 8 different ways that each worm can be positioned and there are 4 different worms, and 8 x 4 is 32. Then you must times 32 by 16 as there are 16 places that the worm head could start at, so there are 512 solutions. I also worked out the smallest and the largest possible combination that the worm could possibly lie on here is what I worked
out...
For the smallest the head must start on 34 and go strait up so it would go 34, 24, 14 and then 4 so the total would be 76. For the largest combination you would have to start on 67 and go straight down so it would go , 67, 77, 87 and then 97 so the total would be: 328
Nathan
I know the total amount of solutions is 512 because there are 32 different shapes the worm can make because normally it can make three shapes in every direction so you would get twelve and then you can reverse it all apart from the straight lines because they will just be the same so you get 20. Then you add on the shapes which are squares to get 24 then you have to twist
it up and you get 4 more and then you reverse it to get 32. Finally you need to times 32 by 16 to get 512
Oliver
The way to figure out all the solutions, is to find out how many there are facing one way, to get 4, then times by 2, because you can reverse each shape to get 8. You can also move it around so it is facing different directions so you times it by 4, because it can face 4 different directions, which is 32 solutions then times by 16, which is 512.
Ryan J. & Ryan W. from Imlay Elementary School in the USA sent in this explanation and their workings:
We know that there are 30 possible directions the 4 block worm can go while still being connected properly, while still eating its favorite food in the garden. We also wrote down the sum of the 4 number squares the worm was on, and drew the position so we could check to see if a position was repeated later. The worm head placement we chose was square number 34. The way
we know we found all the worm positions is because we listed the possibilities that the worm can move each time we found a new one.
We also know we found all of them is because we would come up of new positions, and write them down four times, one pointing up, one pointing left, one down, and one right. After we did thirty, we could come up with nothing else.
The way we knew we didn't miss one, is we crossed out all squares that where to far away than 4 blocks from the worm head. Then we made every combination in the area of the non-crossed out squares.
Thank you all for the solutions sent in.
Teachers' Resources
Worms
Imagine a square patch of ground in a garden, and because it's a good garden there are worms around, and they feed from the soil. Well, I've taken an imaginary square and given it the numbers 1 to 100.
You've probably come across number squares like this in maths sometimes at school. So each part of this square piece of ground has a number. But now, suppose that the centre piece of this square is what the worm likes to eat most of all. So I've taken the centre 16 squares and made them a little darker.
You will probably notice that the lowest number in this square is 34 and the highest is 67. Well, that's enough about the ground. What about the worm? The worm will be 4 squares long and to make sure we know which is the head and which is the tail I've coloured the head part differently.
Of course the worm can bend itself in different ways. But I'm going to suggest that on this occasion it has 4 square parts [like 4 multilink] which have to be joined edge to edge in the "normal'' way. Here are three examples that would be O.K., and one which is not because the head is not at the end of the body!
Now, to set up the challenge. With a square in front of you, with the numbers 1 to 100 marked on as shown and the centre 16 squares marked in some special way, place your worm, perhaps made out of multilink, on the square. [Multilink are 2 cm long so get hold of some 2 cm squared paper; the backing paper for some "taky-back'' film is just the right size.] When you place your worm you have to make sure that the head is on one of the centre squares that are shaded differently. The head stays there, but while the worm is eating it wriggles and changes the shape of its body, in as many different ways as it can. Here are just three of the many ways in which the worm can be, and I've placed its head on the number 34.
The challenge is to:-
- Find all the shapes and positions that the worm can be with its head fixed in the one place [in my example, on number 34, but you can choose any of those 16 squares]. Don't forget about the worm's body going North, South, East and West.
- Add up the numbers that the worm is on, that is the numbers that the head and the rest of the body are resting on. [In my first example 34 + 33 + 43 + 53 = 163; 2 nd example gives 34 + 33 + 23 + 13 = 103; 3 rd example gives 34 + 33 + 23 + 22 =112.]
- Find a good way of recording your results
- so that you can make sure you do not do the same one twice and
- so that you can compare the results easily.
Well, that's really it! Of course at the end you have to say "I wonder what would happen if I ...?''.
Why do this problem?
Possible approach
Key questions
Possible extension
