# Problematics: Tennis in Wonderland

## Why Lewis Carroll might have questioned if Roger Federer deserves his second prize at Wimbledon.

How Lewis Carroll might have rated the victories of Sania-Martina, Djokovic and Federer. (Source: Reuters & AP photographs)

Lewis Carroll wouldn’t have minded Sania Mirza and Martina Hingis winning the Wimbledon women’s doubles. Or Novak Djokovic winning the men’s singles. It is Roger Federer’s second place that Carroll would possibly have questioned. As also the second and third/fourth places in all events of the tournament.

We all know who Carroll was; our childhoods wouldn’t have been the same without ‘Alice in Wonderland’, which incidentally completes 150 years in 2015. You possibly know also that Carroll was a brilliant mathematician — he taught the subject at Christ Church College, Oxford — besides a puzzlist and a logician.

In ‘Lawn Tennis Tournaments’, published in 1883, Carroll analysed tournament formats and found them wanting. The best player would indeed become the champion, he concluded, but the second, third and fourth best players might or might not end up in the positions they deserved.

Carroll’s discussion is based on a field of 32 players. To simplify matter, I have restricted the field to 16 players. In my illustration, the numbers 1 to 16 stand for the relative strengths of the players as shown. As you can see, the fifth best player (Y) wins second position, doing better than the second, third and fourth best players. This is because No.5 is in a different half while No.2, No.3 and No.4 suffer in the same half as No.1.

There was no seeding system in those days. Modern organisers will argue that they distribute the players in a manner that No.1 will not play No.2 before playing No.3 or 4 first, and so on. Had Carroll lived long enough to see seedings, he would have pointed out how artificial they are.

In Carroll’s own example of 32 players, the number he assigns to each is “real”. These numbers, he explains, will be in line with the actual positions the players will achieve in the tournament, although they are not supposed to know it when the event begins. No matter how the draw is made, the second best and third best players will always win the trophies they deserve — as long as they play according to the format Carroll proposes. His proposal is very detailed; the following is a condensed version.

(a) If A defeats B, A is obviously superior to B. If B defeats C, then both A and B are superior to C.
(b) No player is knocked out on the first defeat. Only when a player has three “superiors” is he eliminated.
(c) On the first day, the 32 players play 16 matches. On the second day, the 16 winners play 8 matches and, simultaneously, the 16 losers play another 8 matches.
(d) In the last described group of 8 matches, anyone who loses is eliminated. That is because he has three superiors — one in the first match, one in the second match, and one who had already defeated his second superior.
(e) This reduces the field to 24. Pair the unbeaten players in one group, those with one superior in another group, and so on. Carry on this way, round after round. The champion will be decided early but the second and third positions will be found later.

For a more elaborate description, read ‘The Complete Works of Lewis Carroll’.

Puzzle#19A: Take the Wimbledon men’s singles, which has 128 players. Match them up in an expanded version of Carroll’s format. Now let them play in the traditional Wimbledon format, which will not recognise Carroll’s findings. What is the probability that the second best player (as per Carroll) will also finish second in Wimbledon?

What you wrote
Because of last week’s special episode, we had to postpone the answers of the previous week’s puzzles. So let’s go back two weeks to the week starting July 1, when I gave you Mann ki Baat from two towers with two and eight loudspeakers. And let’s welcome a first-time reader as he seeks to locate the points where the sound intensity from both sources is the same.

Kabir, in Puzzle#17A, intensity drops with the square of distance. So, the point where the intensities (loudness) is the same will be such that the ratio of the distances is √(2:8) or 1:2, being nearer the tower with two speakers. There are obviously two such points — from basic coordinate geometry, (1:2) and (-1:2). The first point is between the towers at a distance 1/6 km from the tower with two speakers. The second point (-1:2) will be 1/2 km from the two-speaker tower and 1 km from 8-speaker tower.

— Phani Bhushan Tholeti (Synaptics biometrics division, Hyderabad)

Sanjay Gupta (New Delhi), M Natrajan (IIM Calcutta alumnus) and Sathya Prakash (New Jersey) identify the same two points as Phani does. Biren Parmar (Texas A&M University) and Sampath Kumar V (IIM Kozhikode alumnus) identify an infinity of such points.

If you stay within the shaded circle, the sound intensity from one tower is higher; if you go outside, the other tower will bring you the louder sound. Along the circumference, you get an equally intense sound from either tower.
As for Puzzle#17B, the longest possible word with Y as the only vowel is RHYTHMS, as echoed by Sampath Kumar V, Phani Bhushan Tholethi, M Natrajan and Sathya Prakash. Anindita Basu (IBM India, Allahabad) forgets to add the S for the plural. Biren Parmar suggests SYZYGY which, though one letter shorter than RHYTHMS, is worthy of mention because Lewis Carroll invented a game called SYZYGIES.

Tennis quiz
Puzzle#19B(i): He romanced a singer 28 years his elder, married an actress and is now married to a tennis player. In one Wimbledon final, he defeated an opponent with a powerful serve. Identify all five characters.
Puzzle#19B(ii): She was world number one until her rival’s fan stabbed her on a tennis court. Though she recovered, she never regained her original position. Name both players and the stabber.
Puzzle#19B(iii): She lost a Wimbledon final and wept on the shoulder of royalty. Identify the loser and the royalty.