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In Particular

Write 100 as the sum of two positive integers, one divisible by 7 and the other divisible by 11. Then find formulas giving all the solutions to 7x + 11y = 100 where x and y are integers.

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For What?

Prove that if the integer n is divisible by 4 then it can be written as the difference of two squares.

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Ordered Sums

Let a(n) be the number of ways of expressing the integer n as an ordered sum of 1's and 2's. Let b(n) be the number of ways of expressing n as an ordered sum of integers greater than 1. (i) Calculate a(n) and b(n) for n<8. What do you notice about these sequences? (ii) Find a relation between a(p) and b(q). (iii) Prove your conjectures.

Training Schedule

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem?

When we watch sports coverage of the multi-discipline events such as the heptathlon, the scoring mechanism is not usually made explicit. This problem explores some of the maths behind the scoring system and invites students to optimise an athlete's performance by choosing a suitable training schedule. Along the way, students can practise substituting into formulas, make sense of functions, and use spreadsheets to repeat routine calculations quickly.

Possible approach

Set the scene by introducing the seven heptathlon events (perhaps asking students if they can name the events). Then display the two equations:
$y=a(b-x)^c$      (1)
$y=a(x-b)^c$      (2) 
"These are the equations used to calculate the points scored in the different heptathlon events. Why do you think there are two equations? Which equation do you think is used for each event?"
Give students time to discuss their answers in pairs, and then share as a class.
Now display the rest of the problem and/or hand out this worksheet. Before they get started, take some time to talk through the problem, and check students understand the task.
As there is quite a bit of repeated computation involved, it may be useful to use spreadsheets if computers are available. Alternatively, students could work in small groups and share out the calculations among themselves.
Finally, gather the class together to share the different training schedules they have devised.  

Key questions

What happens to the number of points given by each equation as $x$ increases/decreases?

Possible extension

Imagine the heptathlon was to become an octathlon. Choose an eighth event and design and justify a scoring formula which would allocate points consistent with the other events.

Possible support

Students may opt to use a calculator to solve this problem, but it is much more efficient to use a spreadsheet. In order to make the most of the task, it may be worthwhile spending some time with the whole class talking about how to set up formulas in a spreadsheet and how to make changes to investigate different training schedules.