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Round Table: Teachers' Notes

Introduction

This problem is well known under a few names. For example, King Arthur’s Problem, and Knights of the Round Table. More details on this problem can be found on the web by using your search engine.
 
The Round Table Bright Sparks activity gives the student a chance to analyse the game and find the winning seat number not only for some specific values of the number of people seated at the table but also for a general number. To find the general answer, students will need to experiment until the game is understood. This provides an opportunity for systematic experimentation and the application of logical reasoning. There is also the opportunity for a student to prove that the general situation that they have conjectured is actually correct. Towards the end we suggest some extensions of the problem that your students might like to try.
 
These notes are not intended to be a sequential description of how to use this activity with students but they should provide you with further information that may help you to better understand the problem and how it might be used to provide a rich mathematical experience for your students. You should also read the Students’ notes for this problem, as they provide specific hints on completing the Bright Sparks activity online.
 

About the problem

Hopefully the problem is clear. You need to keep the first seat and delete the second; keep the third and delete the fourth and so on. Things get a little more difficult after the first round because only the odd numbers are left by then but the same principle applies: keep the next and delete the one after. It should just be a matter of moving around the table in a systematic fashion.

Some possible solutions

Let N be the number of seats and M be the winning seat number. As is often the case in these problems it is always a good idea to build up a table of values. Once the students have done that there is a good chance that they might see some patterns. Here we have listed the first 20 values of N.

N
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
M
1
1
3
1
3
5
7
1
3
5
7
9
11
13
15
1
3
5
7
9

Well, maybe the pattern is pretty clear by now but it may not be so easy to see how to write it down. So before we make the grand conjecture for all N, let’s see a few ‘smaller’ patterns. We list these as possible conjectures.
 
Conjecture 1: M can never be even.
Conjecture 2: If N is a power of 2, then M is 1.
Conjecture 3: M = N if N is one less than a power of 2.
Conjecture 4: If N = 2n + i, for 0 ≤ i < 2n, then M = 2i + 1.
 
Proofs of these conjectures are available on the Round Table Proof page.
 
There may be other conjectures here that we have missed but check them out with further values of N to make sure that they are correct for values other than those in the table. (In fact, check our conjectures out too. Does Conjecture 1 hold for N = 21, 22 and 23? Can the same be said for Conjectures 3 and 4? Is Conjecture 2 true for N = 64?) Then try to prove your conjectures.

Extensions

The question now is can we take the problem any further. What Round Table-like problems can we build around this one? It’s very important to have this discussion with students at all levels. Extending problems is a fundamental way of working for mathematicians and students should know that. Let us suggest some related problems.
  • Suppose that instead of keeping one and deleting the next we keep two and delete the next. Given N what is M then?
  • Suppose that instead of keeping one and deleting the next we keep three and delete the next. Given N what is M then?
  • Suppose that instead of keeping one and deleting the next we keep r and delete the next. Given N what is M then?
  • Suppose that instead of keeping one and deleting the next we keep one and delete the next two. Given N what is M then?
  • Suppose that instead of keeping one and deleting the next we keep one and delete the next three. Given N what is M then?
  • Suppose that instead of keeping one and deleting the next we keep one and delete the next s. Given N what is M then?
  • Suppose that instead of keeping one and deleting the next we keep r and delete the next s. Given N what is M then?