2. The Further Mathematics Support Programme Our aim is to increase the uptake of AS and A level Mathematics and Further Mathematics to ensure that more students reach their potential in mathematics. To find out more please visit www.furthermaths.org.uk www.furthermaths.org.uk The FMSP works closely with school/college maths departments to provide professional development opportunities for teachers and maths promotion events for students.

3. A Level Maths & Further Maths  Highly regarded and popular A levels  Facilitating subjects for universities  Many A level and university subjects require maths knowledge  Opens doors to a variety of careers, many of which are well paid  Enjoyable  Challenging  Useful

4. Careers that need maths qualifications  Engineer  Scientist  Teacher  Computer programmer  Games designer  Internet security  Logistics  Doctor  Dentist  Meteorologist  Finance  Accountancy  Business management  Nursing  Veterinary  Sports science  Physical Therapist  Pharmacist  Medical statistician And many more…

5. Josephus Flavius

6. Magic Cards

7. What does 495 mean? What about 3287?

8. Suppose instead of having a number system based on 10s, we used 2s. What would that look like?

9. What would 17 look like?

10. 10001

11. What about 25?

12. 11001

13. In your workbook.

14. Are these true?  1 + 1 = 10  10 + 1 = 11  10 + 11 = 101

15. Binary Sums Key Addition Results for Binary Numbers  1+0=1  1+1=10  1+1+1=11 Key Subtraction Results for Binary Numbers  1–0=1  10–1=1  11–1=10

16. Binary Sums QuestionAnswer 111+100 101+110 1111+111 111-101 110-11 1100-101 1110+10111 1110+1111 11111+11101

17. Answers QuestionAnswer 111+1001011 101+1101011 1111+11110110 111-10110 110-1111 1100-101111 1110+10111100101 1110+111111101 11111+11101111100

18. Josephus Flavius… Jewish historian 1 century AD Trapped with 40 soldiers Preferring suicide to capture they decided to kill themselves They formed a circle and starting from one, every remaining alternate person was eliminated… Josephus, not keen to die, quickly found the safe spot in the circle and thus stayed alive.

19. An example  We will start with a circle of 10 people.  The arrow in the diagrams will show the person to be left alive at that stage.

30. Conclusions If we always start with 1 then even numbers will not survive! The survivor for a group of 10 people is person number 5. We shall say J(10) = 5.

31. The Challenge  Can you quickly determine J(41), the position of the survivor in a group of 41 people?  What about groups of any size?

32. Complete the table Number of people (n) 123456789 Who wins? J(n) Number of people (n) 101112131415161741? Who wins? J(n)

33. Results Number of people (n) 123456789 Who wins? J(n) 113135713 Number of people (n) 101112131415161741? Who wins? J(n) 5791113151319

34. In Binary n 1111 2 10 3 11 4 100 5 101 6 110 7 111 8 1000 9 1001 J(n)1011100101110111100010011 n 10 1010 11 1011 12 1100 13 1101 14 1110 15 1111 16 10000 17 10001 J(n) 010101111001101111011111 00001 00011

35. 1101 n J(n)

36. 1101 n J(n)

37. 1 n 110

38. n J(n) 110 1 1

39. 1 n J(n) 110 1

40. What is actually happening when we are moving the first digit to the end?

41. Consider what is happening when 495 changes to 954.  First we subtract 400, to give us 95.  Then we multiply by 10, to give us 950.  Then we add 4 to give 954.

42. What is happening in binary?  Consider 1011.  Moving the first digit to the end gives us 0111:  First subtract the highest power of two (removing the one in the far left column, which is worth 8).  Then multiply by two, to move everything left one place value column.  Then add 1.

43. When there were 41 people, Where did Josephus stand?  41-32 = 9 (subtract the highest power of 2)  9x2 = 18 (multiply by 2)  18+1 =19 (add one)  Stand in place 19.

44. In summary! There are 10 types of people in the world; those who understand binary and those who don’t.

45. A Level Maths & Further Maths  Highly regarded and popular A levels  Facilitating subjects for universities  Many A level and university subjects require maths knowledge  Opens doors to a variety of careers, many of which are well paid  Enjoyable  Challenging  Useful

46. Careers that need maths qualifications  Engineer  Scientist  Teacher  Computer programmer  Games designer  Internet security  Logistics  Doctor  Dentist  Meteorologist  Finance  Accountancy  Business management  Nursing  Veterinary  Sports science  Physical Therapist  Pharmacist  Medical statistician And many more…

47. The Further Mathematics Support Programme Our aim is to increase the uptake of AS and A level Mathematics and Further Mathematics to ensure that more students reach their potential in mathematics. To find out more please visit www.furthermaths.org.uk www.furthermaths.org.uk The FMSP works closely with school/college maths departments to provide professional development opportunities for teachers and maths promotion events for students.