Experimenting and Discovery in Mathematics

1. Experimenting and Discovery in Mathematics The Benefits of Computers

2. Sudoku Story

3. Sudoku Story

4. Five Button Door Lock Marc Dostie, on my book’s Facebook page, asks: How Many Combinations?

5. Number of Combinations – First Try: D(n) Order Not Considered No Simultaneous Presses D(1) = 2 D(n) = 2*D(n-1) D(n) = 2n

6. Number of Combinations – Second Try: E(n) Order Considered No Simultaneous Presses E(n) = C(n,0)0! +C(n,1)1! +… + C(n,n)n! E(1) = 2 E(5) = 1*1 + 5*1 + 10*2 + 10*6 + 5*24 + 1*120 = 326

7. Number of Combinations with Order but no Simultaneous Presses – E(n) Recursive Structure for E(n)?

8. Recursive Structure for E(n) E(n) = n*E(n-1) + 1 ≈ n! * e

9. Number of Combinations – Third Try: F(n) F(n) = Number of Ordered Combinations withSimultaneous Presses Allowed Choices and Combos A choice is a subset of n numbers to be pressed simultaneously. There are 2n choices. For example, (123) or 2 or (1345) are choices. A combo is an ordered list of choices that contains at most one of each number from 1 to n. For example, 1 2 3 (45) or 3 (12) 4 are combos.

10. Ad-Hoc Solution for F(5) How many total buttons kout of 5 will be pressed? And, for each k, How many ways to press the buttons in order with simultaneous presses? K = 0 : C(5,0) * 1 = 1 K = 1: C(5,1) * 1 = 5 K = 2: C(5,2) * (1 + 2) = 10 + 20 = 30 Either you press both buttons at once (1) or each separately (2) K = 3: C(5,3) * (1 + 6 + 6) =  10 + 60 + 60 = 130 All three at once (1), or split them 21 or 12,  (3 + 3), or all three separately (6).

11. Ad-Hoc Solution for F(5) K = 4: C(5,4) * (1 + 8 + 6 +  36 + 24) = 375 All at once (1), split 13 or 31 (4 + 4), or split 22 (6), or split 112 or 121 or 211 (12*3), or all separately (24). K = 5: C(5,5) * (1 + 10 + 20 + 60 + 90 + 240 + 120)  =  541 All at once (1), split 14 or 41 (5+5), 23 or 32 (10+10), 113 or 131 or 311 (20+20+20), or 122 or 212 or 221 (30+30+30), or 2111 or 1211 or 1121 or 1112 (60*4), or all separately (120). Total = 541 + 375 + 130 + 30 + 5 + 1 = 1082

12. Generating Combos for F(5) with a Computer Order the choices lexicographically (alphabetically) in this way: (), 1, (12), (123), (1234), (12345), (1235), (124), (1245), (125), (13), (134), (1345), (135), (14), (145), (15), 2, (23),(234), (2345), (235), (24), (245), (25), 3, (34), (345), (35), 4, (45), 5 Algorithm for Generating Combos Start with the blank combo (). Repeat until no new combos: 1. If numbers 1-5 are not all included, add the “first” choice that does not duplicate a number. 2. If all 5 numbers are included, then remove a choice from the right end, and replace it with the “first” choice that does not duplicate a number in the combo, or repeat a previously generated combo.

13. Generating Combos Let’s Try the Algorithm Here are the choices again in order: (), 1, (12), (123), (1234), (12345), (1235), (124), (1245), (125), (13), (134), (1345), (135), (14), (145), (15), 2, (23),(234), (2345), (235), (24), (245), (25), 3, (34), (345), (35), 4, (45), 5 And, the combos…

14. Combos for Smaller Numbers of Buttons

15. Recursive Formula for F(n) F(n) = F(5) = C(5,5)F(0) + C(5,4)F(1) + C(5,3)F(2) + C(5,2)F(3) + C(5,1)F(4) + 1 = 1*1 + 5*2 + 10*6 + 10*26 + 5*150 + 1 = 1082

16. F(n) Combos for 0 ≤ n ≤ 5 Closed Form? Open Question

17. Counting by Number of Steps A step is one simultaneous press of button(s). Tim Woodcock’s idea: Count the combos by the number of steps each uses.

18. More Structure – f(n,k) Let f(n,k) be the number of combos in an n-button lock using k steps. That is, F(n) = f(n,0)+f(n,1)+… +f(n,n)

19. Recursive Structure for f(n,k) f(n,k)=(k+1)*f(n-1,k)+k*f(n-1,k-1) Consider f(5,k) 1. No button 5: f(4,k) 2. Yes button 5 a. Pressed alone: k*f(4,k-1) b. Pressed with other buttons: k*f(4,k)

20. Using Computers in Mathematical Research