Stepping Stone Game

1. Stepping Stone Game Pascal’s Triangle: The Stepping Stone Game How many different routes are there from the Start stone to the Finish stone? Rules: You can only walk East or South from any stone. We will start by looking at 5 possible routes (be careful how you walk)

2. Pascal’s Triangle: The Stepping Stone Game How many routes are there to: 1 1 1 1 1 1 2 1 1 1

3. Pascal’s Triangle: The Stepping Stone Game How many routes are there to: 1 1 1 1 1 3 1 2 1 1 1

4. Pascal’s Triangle: The Stepping Stone Game How many routes are there to: 1 1 1 1 1 3 1 2 3 1 1 1

5. Pascal’s Triangle: The Stepping Stone Game How many routes are there to: 1 1 1 1 1 Can you see all 6 of the routes? 3 1 2 How could you have calculated the 6 routes without the need to draw or visualise them? 6 3 1 1 1

6. 3 routes to this stone Why must there be 6 routes to here? 3 routes to this stone Pascal’s Triangle: The Stepping Stone Game How many routes are there to: 1 1 1 1 1 Can you see all 6 of the routes? 3 1 2 How could you have calculated the 6 routes without the need to draw or visualise them? 6 3 1 What do you have to do to get the number of routes to any stone? 1 1

7. Pascal’s Triangle: The Stepping Stone Game How many routes are there to: 1 1 1 1 1 Can you see all 6 of the routes? 3 1 2 4 5 How could you have calculated the 6 routes without the need to draw or visualising them? 6 15 3 10 1 What do you have to do to get the number of routes to any stone? 35 4 1 10 20 Calculate the total number of routes to the finish stone. 5 35 1 15 70

8. Tetrahedral numbers Triangular numbers Counting numbers Square base Pyramid numbers 1 5 14 30 Pascal’s Triangle: The Stepping Stone Game The numbers are symmetrical about the diagonal line. 1 1 1 1 1 Do you notice anything about the numbers produced by the routes through to the finish stone? 3 1 2 4 5 6 15 3 10 1 35 4 1 10 20 5 35 1 15 70

9. Pascal’s Triangle 1 1 1 1 1 1 1 1 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 1 21 35 35 21 7 1 1 8 8 28 56 70 56 28 1 1 36 9 36 84 9 84 126 126 1 1 10 120 45 45 210 210 10 120 252 1 1 330 165 55 462 330 55 165 11 462 11 1 66 12 220 12 792 66 792 924 1 495 495 220 1 715 286 13 1 78 286 13 715 78 1287 1716 1287 1716 Pascal’s Triangle 1. Complete the rest of the triangle. 1 =20 R0 2. Find the sum of each row. 2 =21 R1 3. Write the sum as a power of 2. 2 4 =22 R2 3 3 R3 8 =23 Counting/Natural Numbers 4 6 4 R4 16 =24 Blaisé Pascal (1623-1662) R5 32 =25 Triangular Numbers 64 =26 R6 =27 128 R7 256 =28 R8 Tetrahedral Numbers 512 =29 R9 1024 =210 R10 2048 =211 R11 4096 =212 R12 Pyramid Numbers (square base) R13 8192 =213

10. 5-a-side 1 1 1 1 2 1 1 1 3 3 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 1 21 35 35 21 7 1 1 8 8 28 56 70 56 28 1 1 36 9 36 84 9 84 126 126 1 1 10 120 45 45 210 210 10 120 252 Choose 0 1 1 330 165 55 462 330 55 165 11 462 11 1 66 12 220 12 792 66 792 924 1 495 495 220 1 715 286 13 1 78 286 13 715 78 1287 1716 1287 1716 R0 nCr R1 R2 R3 In how many ways can a 5-a-side team be chosen from a squad of 10 players? R4 R5 R6 The probability of choosing one particular combination of 5 players is 1/252 252 R7 R8 10C5 R9 R10 R11 R12 R13

11. 1 1 1 1 2 1 1 1 3 3 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 1 21 35 35 21 7 1 1 8 8 28 56 70 56 28 1 1 36 9 36 84 9 84 126 126 1 1 10 120 45 45 210 210 10 120 252 1 1 330 165 55 462 330 55 165 11 462 11 1 66 12 220 12 792 66 792 924 1 495 495 220 1 715 286 13 1 78 286 13 715 78 1287 1716 1287 1716 Remember: The top row is Row 0 Use Pascal’s triangle to determine the number of combinations for each of the following selections.

12. Mix 1 9 1 2 3 1 1 7 4 5 6 1 2 1 8 1 1 3 3 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 1 21 35 35 21 7 1 1 8 8 28 56 70 56 28 1 1 36 9 36 84 9 84 126 126 1 1 10 120 45 45 210 210 10 120 252 1 1 330 165 55 462 330 55 165 11 462 11 1 66 12 220 12 792 66 792 924 1 495 495 220 1 715 286 13 1 78 286 13 715 78 1287 1716 1287 1716 C A Choose 7 cards Choose 3 books D B Choose 4 balls Choose 5 players

13. Lottery 14 10 11 13 12 15 1 1 4 2 3 5 6 1 1 16 17 18 19 9 7 1 2 1 1 1 3 3 34 30 31 33 32 35 24 20 21 23 22 25 1 4 6 4 1 36 37 38 39 26 27 28 29 1 5 10 10 5 1 44 40 41 43 42 45 1 6 15 20 15 6 1 1 7 1 21 35 35 21 7 46 47 48 49 8 1 1 8 8 28 56 70 56 28 1 1 36 9 36 84 9 84 126 126 1 1 10 120 45 45 210 210 10 120 252 1 1 330 165 55 462 330 55 165 11 462 11 1 66 12 220 12 792 66 792 924 1 495 495 220 1 715 286 13 1 78 286 13 715 78 1287 1716 1287 1716 49 balls choose 6 National Lottery Jackpot? ?

14. 14 10 11 13 12 15 1 4 2 3 5 6 16 17 18 19 9 7 34 30 31 33 32 35 24 20 21 23 22 25 36 37 38 39 26 27 28 29 49C6 44 40 41 43 42 45 46 47 48 49 8 There are 13 983 816 ways of choosing 6 balls from a set of 49. So buying a single ticket means that the probability of a win is 1/13 983 816 Choose 6 13 983 816 Row 49 Row 0 49 balls choose 6 National Lottery Jackpot?

15. Pierre de Fermat (1601 – 1675) Historical Note Historical Note Pascal was a French mathematician whose contemporaries and fellow countrymen included Fermat, Descartes and Mersenne. Among his many achievements was the construction of a mechanical calculating machine to help his father with his business. It was able to add and subtract only, but it was a milestone on the road to the age of computers. He corresponded with Fermat on problems that led to the new branch of mathematics called Probability Theory. The two problems that they examined concerned outcomes when throwing dice and how to divide the stake fairly amongst a group of players if a game was interrupted. These investigations led Pascal to construct tables of probabilities that eventually led to the triangle of probabilities that bears his name. Blaisé Pascal (1623-1662)

16. Worksheet 1 Pascal’s Triangle: The Stepping Stone Game

17. Worksheet 2

18. Worksheet 3 1 1 1 1 2 1 1 1 3 3 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 1 21 35 35 21 7 1 1 8 8 28 56 70 56 28 1 1 36 9 36 84 9 84 126 126 1 1 10 120 45 45 210 210 10 120 252 1 1 330 165 55 462 330 55 165 11 462 11 1 66 12 220 12 792 66 792 924 1 495 495 220 1 715 286 13 1 78 286 13 715 78 1287 1716 1287 1716