Open squares
Problem
Here are the first five open squares:
(Note that although they are all called 'open squares' the first two are not actually open, but we will still use them in these challenges.)
What would the next five open squares look like?
How many small cubes would each one use?
CHALLENGE 1
You can only use one of any size open square in each total you make, for example 16 + 36 = 52 is right but 4 + 4 + 8 + 36 = 52 is not allowed because it used two 4s.
If there are some totals between 50 and 60 you cannot make, explain why that is so.
CHALLENGE 2
We now stack the open squares up to make hollow pyramids, each open square resting on the one that is the next size bigger.
We can use just the first six hollow pyramids. Here are the first five, you will also need to consider the sixth:
This time your challenge is to use one, two, three or four hollow pyramids and to find the total number of cubes used. However, you cannot use two of the same pyramid in any total.
Can you find a way to make all totals up to and including 59 small cubes?
If there are some totals you cannot make, can you explain why?
Student Solutions
If we consider this, then, each time, except the first term, is a pattern of increasing by four. Or you could say for any term (nth term) the side length of the shape of that term is going to be the same number as the term number, for example, for 6th term, the side length of that term's shape is going to be 6 by 6, and the hollow part in the middle is going to be 4 by 4, therefore, we can also conclude that any one opens square's outer sides and inner sides would always differ by 2.
For example, for 6th term, the side length of that term's shape is going to be 6 by 6, and the hollow part in the middle is going to be 4 by 4, therefore, the number of cubes is 6 - 4 = 2.
3rd term: outer side; 3 by 3, inner side; 1 by 1 (total cubes 8, because, 9 - 1 = 8)
4th term: outer side; 4 by 4, inner side; 2 by 2 (total cubes 12, because, 16 - 4 = 12)
5th term: outer side; 5 by 5, inner side; 3 by 3 (total cubes 16, because, 25 - 9 = 16)
6th term: outer side; 6 by 6, inner side; 4 by 4 (total cubes 20, because, 36 - 16 = 20)
7th term: outer side; 7 by 7, inner side; 5 by 5 (total cubes 24, because, 49 - 25 = 24)
8th term: outer side; 8 by 8, inner side; 6 by 6 (total cubes 28, because, 64 - 36 = 28)
9th term: outer side; 9 by 9, inner side; 7 by 7 (total cubes 32, because, 81 - 49 = 32)
10th term: outer side; 10 by 10, inner side; 8 by 8 (total cubes 36, because, 100 - 64 = 36)
CHALLENGE 1 SOLUTION
So first what we need to do is find the total of each of the 10 squares, or we can simply find the nth term. For example, the first square would be four little cubes. After we have done this, we need to find either 2, 3, or 4 of these squares so that the total number of small cubes will add up to a number between 50 and 60.
The pattern is, 1, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, however, we only need these numbers: 1, 4, 8, 12, 16, 20, 24, 28, 32, 36 so we can choose any 2, 3 or 4 of these numbers that will add up to a number between 50 and 60, remember, not including 50 or 60.
This would take some time, as you need to work out different combinations that work. The following are some of the combinations that work:
1,4,12,36 = 53
1,4,16,32 = 53
1,4,16,36 = 57
1,4,20,28 = 57
1,4,20,32 = 57
1,4,24,28 = 57
16,12,28 = 56
16,36 = 52
20,32 = 52
20,36 = 56
24,28 = 52
24,32 = 56
I think that there are some totals between 50-60 you cannot make, because some totals need repeated numbers, however, that is not allowed, such as, 59.
Finally, some numbers can't be made, because there could only be 2, 3, 4 numbers used, therefore, you can't use more nor less, however, some numbers require five different numbers added together.
Thank you for your thoughtful response, Frank. I wonder whether anyone has had chance to think about the pyramids in the second challenge?
Teachers' Resources
Open Squares
Here are the first five open squares:
(Note that although they are all called 'open squares' the first two are not actually open, but we will still use them in these challenges.)
What would the next five open squares look like?
How many small cubes would each one use?
CHALLENGE 1
You can only use one of any size open square in each total you make, for example 16 + 36 = 52 is right but 4 + 4 + 8 + 36 = 52 is not allowed because it used two 4s.
If there are some totals between 50 and 60 you cannot make, explain why that is so.
CHALLENGE 2
We now stack the open squares up to make hollow pyramids, each open square resting on the one that is the next size bigger.
We can use just the first six hollow pyramids. Here are the first five, you will also need to consider the sixth:
This time your challenge is to use one, two, three or four hollow pyramids and to find the total number of cubes used. However, you cannot use two of the same pyramid in any total.
Can you find a way to make all totals up to and including 59 small cubes?
If there are some totals you cannot make, can you explain why?
Why do this problem?
This activity was designed for the 2016 National Young Mathematicians' Award (NYMA) so it might be a particularly useful activity for a small group of your highest-attaining pupils to work on. (For more information about the NYMA, see this brief article which is part of this feature.) This task is a useful vehicle for developing systematic approaches. It can be used as an activity to encourage children to explain in written or spoken words what it is they have done.
Possible approach
Since this activity is aimed at the most confident mathematicians, there will not be much that you have to do to introduce it apart, perhaps, from allowing access to many inter-connecting cubes.
You may find this double-sided sheet of the challenge useful for printing and giving out to learners: pdf or Word document
Key questions
Tell me about how you are getting a solution for the challenge you are working on.
(When encouraging the pupils as they work on the task, try to avoid saying things about what you notice and directing them in your way of attempting a solution.)
Possible extension
Learners might like to try the other two challenges that were part of the National Young Mathematicians' Award 2016: Dart Target and Painting Possibilities.