1. Tennis is Surreal? Joe McCarry Laredo Community College Texas Section MAA, 2016 at Stephen F. Austin State University

2. Make a choice. Tennis counts Love, 15,30,40,deuce,add-in,add-out,… I decided to go with something even more complicated.

3. Berlekamp’s (Hackenbush) method associates a value…as follows. AMS Feature Column on Combinatoric Games http://www.ams.org/samplings/feature-column/fcarc- partizan3#sthash.zeDI36D0.dpufhttp://www.ams.org/samplings/feature-column/fcarc- partizan3#sthash.zeDI36D0.dpuf

4. Berlekamp’s method 1.The sign of the answer depends on whether the edge attached to the green line is red or blue. If the attaching edge is blue, the value is positive and if the attaching edge is red, the value is negative. 2.The value is an integer if the whole string is one color.

5. Blue Wins (no sign change) +4 BlueRedBerlekamp Sequence Berlekamp Score LL 15L1+1 30L1+1 40L1+1 GL1 =+4

6. Berlekamp’s method 1.The sign of the answer depends on whether the edge attached to the green line is red or blue. If the attaching edge is blue, the value is positive and if the attaching edge is red, the value is negative. 2.The value is an integer if the whole string is one color. 3.Otherwise there is a place where the color changes from red to blue or blue to red. This pair of consecutive edges is interpreted as a "binary point“. 4.The additional value of the edges beyond this pair is coded in binary, blue edges being assigned a 1 and red edges a 0. 5.At the very end, one terminates this sequence by adding a 1.

7. Blue wins a game. +11/4=2.75 BlueRedSequenceBerlekamp LL1+1/4 L150-1/2 L300binary 15301point 30 1+1 40301+1 G1 +3.01=+3-1/4

8. Choice I decided that tennis requires me to start my sequence with the last point of the game.

9. If you start the sequence at the other end you get a negative value(-1/16) implying red won. BlueRedSequenceBerlekamp LL L150 L300binary 15301point 30 1+1/2 40301+1/4 G1+1/8 1+1/16 -1.1111=-1+15/16

10. Choice I decided that tennis requires me to start my sequence with the last point of the game. The winning point determines the sign of the game, as it should. I like this choice for tennis because it values late points more than earlier points. It values “momentum” at the end of a game.

11. Blue wins a game. +15/16=.9375 BlueRedSequenceBerlekamp LL1+1/16 15L1+1/8 30L1+1/4 30150-1/2 30 0binary 40301point G1+1 1.0111=+1-1/16

12. Red wins a game. -14/16=-.875 BlueRedSequenceBerlekamp LL+1/16 L150-1/8 L300-1/4 15301+1/2 30 1binary 30400point G0 =0.1001=-1+2/16

13. I don’t like that Blue’s score was +15/16 Red’s score was -14/16 For the mirror image game. Make another choice.

14. Berlekamp’s method 1.The sign of the answer depends on whether the edge attached to the green line is red or blue. If the attaching edge is blue, the value is positive and if the attaching edge is red, the value is negative. 2.The value is an integer if the whole string is one color. 3.Otherwise there is a place where the color changes from red to blue or blue to red. This pair of consecutive edges is interpreted as a "binary point“. 4.The additional value of the edges beyond this pair is coded in binary, blue edges being assigned a 1 and red edges a 0. 5.At the very end, one terminates this sequence by adding a 1.

15. A game with a deuce. +43/32=+1.34375 BlueRedSequenceBerlekamp LL 15L1+1/32 15 0-1/16 30151+1/8 30 0-1/4 40301+1/2 deuce 0binary Add-inAdd-out1point G1+1 +1.10101=+1+11/32

16. Tennis games go on forever if players alternate points. BlueRedSequenceBerlekamp LL 15L1 0 30151 30 0 40301 deuce 0 Add-inAdd-out1 deuce 0 Add-inAdd-out1 deuce 0

17. Once you are into deuces, you can at any point splice in a new Ad-in or Ad-out sequence. BlueRedSequenceBlueRed LLLL 15L11 L 00 3015113015 30 00 4030114030 deuce 00 Add-inAdd-out10 Add-in deuce 01 Add-inAdd-out11Add-inAdd-out deuce 00

18. Once you are into deuces, you can at any point splice in a new Ad-in or Ad-out sequence. BlueRedBlueRed LLLL 15L L 30153015 30 40304030 deuce Add-inAdd-out Add-in deuce Add-outAdd-in Add-out deuce Add-inAdd-outAdd-inAdd-out

19. The lower bound for a blue win (finite game) would have to end with two blue points. 1,1,0,0,1,0,1,0… 1.010101010… +1-1/2+1/4-1/8… +1-(1/2-1/4+1/8…) +1 -1/3 2/3

20. The upper bound for a red win (finite game) would have to end with two red points. 0,0,1,1,0,1,0,1,0,1,0… -1.10101010… -1+1/2-1/4+1/8… -1+(1/2-1/4+1/8…) -1 +1/3 -2/3

21. Types of Tennis Games SmallFiniteBoundInfinite Surreal Live Here? -0+0Infinite Surreals Live Here? BoundFiniteBig -4-3,-5/2,- 3/2 -2/30,1,0,1 …1,0,0, 1,…. 0,1,0,1 …. 1,0,1,0 …. 1,0,1,0 …0,1,1, 0,1,0…. 2/33/2,5/2, 3 4

22. Types of Tennis Games SmallFiniteBoundInfinite Surreals live here? -0Cut+0Infinite Surreals Live Here? BoundFiniteBig -4-3,-5/2,- 3/2 -2/30,1,0,1 …1,0,0, 1,…. 0,1,0,1 …. 01,0,1,0 …. 1,0,1,0 …0,1,1, 0,1,0…. 2/33/2,5/2, 3 4

23. Endless (Surreal)