TS-9.3

teachwire.net/secondary GOING DEEPER Confident students could explore what happens if you miss out two people, instead of one person, each time as you go round the circle. MAIN ACTIVITY ABOUTOUR EXPERT Colin Foster is a Reader in Mathematics Education at the Mathematics Education Centre at Loughborough University. He has written many books and articles for mathematics teachers. His website is foster77.co.uk , and on Twitter he is @colinfoster77 ADDITIONAL RESOURCE An interactive version of the problem (but starting by deleting person 2, rather than person 1) is at bit.ly/geo-josephus 71 Q Why should it be person 4 who wins? How does the 4 depend on the total number of people? Try this counting- out game with different numbers of people and see what happens. Students may come up with wild ideas about possible connections between 10 and 4 (subtract 6, halve it and add 1, etc.). This is fine, as these will be contradicted very soon as students gather more data and gain more insight into the problem. Generating data systematically reveals the key role of powers of 2. Whenever there are 2 m people, the winner is always the 2 m th person. Students could explore what happens when the number of people is, say, one more or one fewer than this. Students could start by acting out the process with real students sitting in a circle. A tap on the shoulder means that they are ‘out’. If they collect their results in a table, they will notice that the pattern in the answers seems to be multiples of 2, which resets every time we hit a power of 2: Total number of people Last person standing Patterns noticed 1 1 2 0 2 2 2 1 3 2 2 4 4 2 2 5 2 2 6 4 4 7 6 6 8 8 2 3 9 2 2 10 4 4 11 6 6 12 8 8 13 10 10 14 12 12 15 14 14 16 16 2 4 Free Mathematics Support Resources from the Association of Teachers of Mathematics During this uncertain time, while our normal teaching pattern has been disrupted, ATM has made available a selection of Free to Access resources suitable for use within schools and home- learning environments. We will be adding to these resources in the weeks ahead. atm.org.uk/Free- ATM-Resources- DISCUSSION Q What did you find out? How does who wins depend on howmany people there are at the start? Students will notice things like the fact that the odd numbers get eliminated first, and then alternate even numbers, and so on. So the answer will never be an odd-numbered person, except for the case where there is just one person to start with. It is hard to obtain a formula for the winning person, but students may be able to figure out who the winner is, even for quite large numbers, without having to write out all the numbers. For example, you could challenge them to work out which would be the winning number for all the students in the class, without actually doing it. Supposing that there are around 30 students in the class, students could reason as follows: - For 32 students (a power of 2), the winner will be student 32. - For 33 students, it will be student 2. - For 34 students, it will be student 4. ...and so on. Similarly, for 31 students, we will need to calculate 32 – 2 = 30, so it will be student 30. For 30 students, it will be student 32 – 2 – 2 = 28, and so on. Getting a full formula is likely to be too hard, but students may get some way towards expressing this. Let n be the total number of people. Then winner = 2( n – p ), where p is the highest power of 2 less than n . If we want to express p in terms of n as well, we can do so using logarithms and the ceiling function (indicated with [ x ] brackets, which means the closest integer greater than or equal to x ). Using this notation, so the winner is given by A spreadsheet of values is available at teachwire.net/counting-winners p = 2 [ log 2 n ] – 1 winner = 2 ( n – 2 [ log 2 n ] – 1 )