1. THE BENEFITS OF COMPUTERS Experimenting and Discovery in Mathematics

2. Sudoku Story 728 79 86417 9 1428 7 12945 748 861

3. 637159428 451278963 982364517 248715639 316492785 795836241 123947856 574681392 869523174

4. Five Button Door Lock Marc Dostie, on my book’s Facebook page, asks: Facebook page 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) = 2 n

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 with Simultaneous Presses Allowed Choices and Combos A choice is a subset of n numbers to be pressed simultaneously. There are 2 n 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 k out 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… ()11 21 2 31 2 3 41 2 3 4 51 2 3 (45)1 2 3 51 2 3 5 41 2 (34) 1 2 (34) 51 2 (345)1 2 (35)1 2 (35) 41 2 4…

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