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Handicaps

 

One of the beauties of Real Tennis is the handicapping system, which, allows, within reason, any two players, or four in doubles, to have a competitive match.

In most day-to-day matches, the handicap differences mean that one player, the one with the higher (worse) handicap, starts with a slight advantage and/or the lower (better) handicap player starts with a slight disadvantage, in each game.

For example, if the handicap difference is 6, then the higher handicapper will start every game 15-0 up (called "receive 15"), as if they had won the first point, but before a ball has been struck!

As the handicap difference decreases, so does the advantage or disadvantage, with a difference of 1 meaning that the lower handicap player starts every fourth game 0-minus 15 (called "owe quarter 15").

Similarly, as the difference increases, so does the advantage or disadvantage. The maximum difference of 50 (for singles) means that:

  • each game starts "receive 30, owe 40" (ie the lower handicapper must win 7 points just to get to the starting score of the higher handicapper), and
  • the better player has only one serve, and
  • the better player cannot hit the tambour from the service end, and
  • the better player cannot lay chases worse than 2 and 3, and
  • the better player cannot lay hazard chases when playing off a chase, and
  • the better player concedes the service end after one chase!

Pity the poor marker!

A quirk of the system is that when playing off handicap there is no deuce/advantage at 40-40. Instead the nerve-shredding next point wins the game!

This reflects the statistical nature of the system, which uses the likelihood of one or other playing winning a certain number of points. With deuce/advantage this unfairly favours the better player, who, even if they lose a deuce point, will always be more likely to win the next point, when advantage down.

Handicaps range from roughly +15 (usually the world champion!) to 95 (the most beginner of beginners), meaning that a better-than-average club player should be able to play either extreme, and have a competitive match (but a very different experience!).

In reality, most clubs arrange matches between players of broadly similar handicaps, eg typically with a maximum difference of 6 handicap points at the Royal Tennis Court. When the difference is just 1 or 2 handicap points, players will often agree to play off level, which brings deuce and advantage into play.

Handicap-based intra-club competitions can see large handicap-difference matches, often in the latter stages if the entrants are banded/seeded in early rounds. Playing with or against one of the larger handicap differences is a real skill for both players, and one that needs serious practice!

The Real Tennis Online service manages handicaps and match results for all clubs and courts worldwide.

 

Handicap Table

 

This table sets out the handicaps to be applied in both singles and doubles, depending on the difference between the handicaps of the two players/pairs).

An owe handicap means that the better player, with the lower handicap, must win a point or points just to get back to love. A receive handicap means that the worse player, with the higher handicap, starts a game as if having already won a point or points.

A bisque allows the player or pair to 'claim' a point at any stage.

The ban combinations for the much larger handicap differences aim to restrict the better player, forcing them to favour precision over power, and removing some shortcuts to winning points.

Handicap
Difference
Receiving
Player
Owing
Player
   
Singles Doubles     Additional Handicap
1 1 Love Owe ¼ 15 Or a ½ bisque
2 2/3 Love Owe ½ 15 Or a bisque
3 4 Receive ½ 15 Love
4 5 Love Owe 15
5 6 Receive ½ 15 Owe ½ 15
6 7 Receive 15 Love
7 8 Receive ½ 15 Owe 15
8 9 Receive 15 Owe ½ 15
9 10 Receive ½ 15 Owe ½ 30
10 11 Receive 15 Owe 15
11 12 Receive ½ 15 Owe 30
12 13 Receive 15 Owe ½ 30
13 14 Receive 15 Owe 30
14 15/16 Receive ½ 30 Owe 15
15 17/18 Receive ½ 30 Owe ½ 30
16/17 19/20 Receive ½ 30 Owe 30
18/19 1/22 Receive 30 Owe 15
20/21 3/24 Receive 30 Owe ½ 30
22/23 5/26 Receive 30 Owe 30
24/25 7/28 Receive 30 Owe 40
26/27 9/30 Receive 30 Owe 15 One serve & banned tambour
28/29 1/32 Receive 30 Owe ½ 30 As above
30/31 3/34 Receive 30 Owe 30 As above
32/33 5/36 Receive 30 Owe 40 As above
34/35 7-39 Receive 30 Owe 15 As above, & chases worse than 3 & 4. Banned hazard chases when playing off chases & Service End is conceded after one chase
36/37 0-42 Receive 30 Owe ½ 30 As above
38/39 3-45 Receive 30 Owe 30 As above
40/41 6-48 Receive 30 Owe 40 As above
42/43 9-51 Receive 30 Owe 30 As above except banned chases worse than 3
44/45 2-54 Receive 30 Owe 40 As above
46/47 5-57 Receive 30 Owe ½ 30 As above except banned chases worse than 2 & 3
48/49 8-60 Receive 30 Owe 30 As above
50+ 61+ Receive 30 Owe 40 As above
 

This table comes from the original, and definitive, guide to handicaps, available here, published by the Tennis and Rackets Association (T & RA).


How Handicaps Change

 

Even the most experienced of players are at times confused by the apparent black art behind handicap adjustments.

Handicapping is only ever an approximation of relative skill level, whose purpose is to enable competitive play between players of different standards. Match results will naturally vary, such that if A has a better handicap than B, it doesn't mean that player A will always beat player B - but over a number of matches, A would be expected to win more often than not. Playing off handicap evens things up and gives both players scope to prove they are better than their rating.

A handicap difference of 6 points is taken to mean that a game played from a 15-0 start will be equally likely to be won by either player. If those players instead started each game from love all, then the better player would be expected to win 2 out of 3 games. This is the statistical basis of the likelihoods of different outcomes based on actual handicaps and the effective handicap differences played.

The handicap difference between two players is reviewed in light of a match result between them, to decide whether that difference is likely to still be accurate, or to need adjustment. If a change is needed, then each player's handicap is altered by the same amount, one favourably, the other not. Thus your handicap is based upon your whole history of recorded results, and the more results you record, the more accurate it is likely to be.

To determine the size of any handicap adjustment, the system considers the match result in the context of the expected outcome, which depends on any disparity between the players' actual handicap difference and the handicap difference at which the match was played.

The system totals up the games won by each player, adding an extra game for each set won. So if A recorded a score of 6-4, 3-6, 3-5 against B, this would be 12-15 in games, plus an extra point each for the completed sets, so 13-16. So A won 13/29 or 44.8%. (We'll refer to this as the 'result %' and the points as 'outcome points' to distinguish them from handicap points and the points in each game!)

The expected outcome depends on whether the handicap difference played reflected the actual handicap difference. If it does match, then the effective handicap difference is 0, and the expected result is 50% each. If it doesn't match, then the expected result varies according to the difference between actual and played handicaps. If the real handicap difference is 2 points but the match is played off level, then the weaker player is under-compensated by 2 points, and so is expected to win fewer than half of the games. The approximate proportion of games the weaker player is expected to win for each effective handicap difference value is shown in the table below (43.3% for a difference of 2):

Level 1 2 3 4 5 6 7 8 9 10
50% 46.7% 43.3% 39.7% 36.6% 33.3% 29.6% 26.5% 23.1% 19.4% 16%

The statistical challenge lies in how much deviation from the expected result is acceptable before adjustment of the handicaps is required, and then how much adjustment is then needed. Current practice recognises three outcomes:

  • a draw, ie the result matches the expected outcome plus or minus around 8% (so approximately 42-58% for Level)
  • a small win/loss, ie outside the draw range, which moves the handicaps by +/- 0.2
  • a big win/loss, which moves the handicaps by +/- 0.6, and is applied to extreme cases of the above which occur less often than 1 in 10 (result % outside approximately 30%/70% for Level.)

(There is a move to replace the arbitrary distinction between small and big result adjustments with a sliding scale of adjustments ranging from 0.2 to 1.0 according to the scale of the result, but this has been awaiting international handicap committee approval for some time.)

This mechanism requires that the 'better' player wins well enough to justify the handicap difference. It can turn out that though the better player wins the match, it records as a handicap draw, or even a loss, because the winning margin was not sufficient to justify the handicap difference.

NB The approximate percentages given above are for guidance only and are not precise. The system in fact uses a table of threshold values measured in games and sets won, according to the effective handicap difference played (as described above) and the total number of outcome points (as above), which progressively increases the significance of each result according to the number of games played within it. For each combination, a set of threshold values is defined which delimits the ranges of outcome points which constitute a big loss, loss, draw, win or big win.

The table is attached below, and as an example, if there are 14 points for, say, a correctly handicapped 8-5 win, the entry for 0 difference and 14 points shows that 0, 1, 2 or 3 games means a big loss, 4 or 5 a loss, 6, 7 or 8 a draw, 9 or 10 a win, and 11, 12, 13 or 14 a big win. So the 8-5 result, reflected as 9 points to 5, is a win/loss.

The table below shows the handicap thresholds.

Other features which might affect your handicap include the following:

  • Home court advantage, to reflect the relative unfamiliarity of a visiting player (so a visitor's handicap is deemed to be higher than recorded), which only applies to singles matches, and is limited to the first few games on a new court
  • 'Volatile' players, whose handicaps are provisional, ie changing so quickly that special steps are needed to adjust them faster by using stable players as yardsticks (doesn't necessarily mean they are temperamental!)
  • Weightings, to increase the size of handicap adjustments after more important club competition and National League matches
  • Pro adjustment, allowing club professionals to manually disarm bandits improving too fast for the system!

(Copied with permission from the original explanation by Oxford University Tennis Club.)

 

Handicap Thresholds

 
Handicap
Difference
Games
Played
Big
Loss
Loss Draw Win Big
Win
-6 1 0 1 2 2 1000
-6 2 0 1 3 3 1000
-6 3 1 2 3 4 1000
-6 4 1 2 4 5 1000
-6 5 2 3 5 6 1000
-6 6 2 4 6 7 1000
-6 7 3 4 6 8 1000
-6 8 3 5 7 9 1000
-6 9 4 6 8 9 1000
-6 10 5 6 9 10 1000
-6 11 5 7 9 11 1000
-6 12 6 8 10 12 1000
-6 13 6 8 11 13 1000
-6 14 7 9 11 13 1000
-6 15 7 10 12 14 1000
-6 16 8 11 13 15 1000
-6 17 9 11 14 16 1000
-6 18 9 12 14 17 1000
-6 19 10 13 15 17 1000
-6 20 10 13 16 18 1000
-6 21 11 14 17 19 1000
-6 22 12 15 17 20 1000
-6 23 12 15 18 21 1000
-6 24 13 16 19 21 1000
-6 25 14 17 19 22 1000
-6 26 14 17 20 23 1000
-6 27 15 18 21 24 1000
-6 28 15 19 22 24 1000
-6 29 16 19 22 25 1000
-6 30 17 20 23 26 1000
-6 31 17 21 24 27 1000
-6 32 18 21 24 27 1000
-6 33 19 22 25 28 1000
-6 34 19 23 26 29 1000
-6 35 20 23 27 30 1000
-6 36 20 24 27 31 1000
-6 37 21 25 28 31 1000
-6 38 22 25 29 32 1000
-6 39 22 26 29 33 1000
-6 40 23 27 30 34 1000
-6 41 24 28 31 34 1000
-6 42 24 28 32 35 1000
-6 43 25 29 32 36 1000
-6 44 26 30 33 37 1000
-6 45 26 30 34 37 1000
-6 46 27 31 34 38 1000
-6 47 28 32 35 39 1000
-6 48 28 32 36 40 1000
-6 49 29 33 36 40 1000
-6 50 29 34 37 41 1000
-6 51 30 34 38 42 1000
-6 52 31 35 39 43 1000
-6 53 31 36 39 43 1000
-6 54 32 36 40 44 1000
-6 55 33 37 41 45 1000
-6 56 33 38 41 46 1000
-6 57 34 38 42 46 1000
-6 58 35 39 43 47 1000
-6 59 35 40 44 48 1000
-6 60 36 41 44 49 1000
-5 1 0 1 2 2 1000
-5 2 0 1 3 3 1000
-5 3 1 2 3 4 1000
-5 4 1 2 4 5 1000
-5 5 2 3 5 6 1000
-5 6 2 4 6 7 1000
-5 7 3 4 6 8 1000
-5 8 3 5 7 8 1000
-5 9 4 6 8 9 1000
-5 10 4 6 8 10 1000
-5 11 5 7 9 11 1000
-5 12 5 7 10 12 1000
-5 13 6 8 10 12 1000
-5 14 6 9 11 13 1000
-5 15 7 9 12 14 1000
-5 16 8 10 12 15 1000
-5 17 8 11 13 15 1000
-5 18 9 11 14 16 1000
-5 19 9 12 15 17 1000
-5 20 10 13 15 18 1000
-5 21 10 13 16 18 1000
-5 22 11 14 17 19 1000
-5 23 12 15 17 20 1000
-5 24 12 15 18 21 1000
-5 25 13 16 19 21 1000
-5 26 13 16 19 22 1000
-5 27 14 17 20 23 1000
-5 28 15 18 21 24 1000
-5 29 15 18 21 24 1000
-5 30 16 19 22 25 1000
-5 31 16 20 23 26 1000
-5 32 17 20 23 27 1000
-5 33 17 21 24 27 1000
-5 34 18 22 25 28 1000
-5 35 19 22 25 29 1000
-5 36 19 23 26 30 1000
-5 37 20 24 27 30 1000
-5 38 21 24 28 31 1000
-5 39 21 25 28 32 1000
-5 40 22 26 29 32 1000
-5 41 22 26 30 33 1000
-5 42 23 27 30 34 1000
-5 43 24 28 31 35 1000
-5 44 24 28 32 35 1000
-5 45 25 29 32 36 1000
-5 46 25 30 33 37 1000
-5 47 26 30 34 38 1000
-5 48 27 31 34 38 1000
-5 49 27 31 35 39 1000
-5 50 28 32 36 40 1000
-5 51 28 33 36 40 1000
-5 52 29 33 37 41 1000
-5 53 30 34 38 42 1000
-5 54 30 35 38 43 1000
-5 55 31 35 39 43 1000
-5 56 31 36 40 44 1000
-5 57 32 37 40 45 1000
-5 58 33 37 41 45 1000
-5 59 33 38 42 46 1000
-5 60 34 39 42 47 1000
-4 1 0 0 2 2 1000
-4 2 0 1 3 3 1000
-4 3 1 2 3 4 1000
-4 4 1 2 4 5 1000
-4 5 1 3 5 6 1000
-4 6 2 3 5 7 1000
-4 7 2 4 6 7 1000
-4 8 3 5 7 8 1000
-4 9 3 5 7 9 1000
-4 10 4 6 8 10 1000
-4 11 4 6 9 10 1000
-4 12 5 7 9 11 1000
-4 13 5 8 10 12 1000
-4 14 6 8 11 13 1000
-4 15 6 9 11 13 1000
-4 16 7 9 12 14 1000
-4 17 7 10 13 15 1000
-4 18 8 11 13 16 1000
-4 19 9 11 14 16 1000
-4 20 9 12 15 17 1000
-4 21 10 13 15 18 1000
-4 22 10 13 16 19 1000
-4 23 11 14 17 19 1000
-4 24 11 14 17 20 1000
-4 25 12 15 18 21 1000
-4 26 12 16 19 21 1000
-4 27 13 16 19 22 1000
-4 28 14 17 20 23 1000
-4 29 14 17 20 24 1000
-4 30 15 18 21 24 1000
-4 31 15 19 22 25 1000
-4 32 16 19 22 26 1000
-4 33 16 20 23 26 1000
-4 34 17 21 24 27 1000
-4 35 18 21 24 28 1000
-4 36 18 22 25 29 1000
-4 37 19 22 26 29 1000
-4 38 19 23 26 30 1000
-4 39 20 24 27 31 1000
-4 40 20 24 28 31 1000
-4 41 21 25 28 32 1000
-4 42 22 26 29 33 1000
-4 43 22 26 30 33 1000
-4 44 23 27 30 34 1000
-4 45 23 27 31 35 1000
-4 46 24 28 32 36 1000
-4 47 24 29 32 36 1000
-4 48 25 29 33 37 1000
-4 49 26 30 33 38 1000
-4 50 26 31 34 38 1000
-4 51 27 31 35 39 1000
-4 52 27 32 35 40 1000
-4 53 28 32 36 40 1000
-4 54 28 33 37 41 1000
-4 55 29 34 37 42 1000
-4 56 30 34 38 42 1000
-4 57 30 35 39 43 1000
-4 58 31 35 39 44 1000
-4 59 31 36 40 44 1000
-4 60 32 37 41 45 1000
-3 1 0 0 2 2 1000
-3 2 0 1 3 3 1000
-3 3 0 2 3 4 1000
-3 4 1 2 4 5 1000
-3 5 1 3 4 6 1000
-3 6 2 3 5 6 1000
-3 7 2 4 6 7 1000
-3 8 3 4 6 8 1000
-3 9 3 5 7 9 1000
-3 10 3 5 8 9 1000
-3 11 4 6 8 10 1000
-3 12 4 7 9 11 1000
-3 13 5 7 10 12 1000
-3 14 5 8 10 12 1000
-3 15 6 8 11 13 1000
-3 16 6 9 11 14 1000
-3 17 7 9 12 14 1000
-3 18 7 10 13 15 1000
-3 19 8 11 13 16 1000
-3 20 8 11 14 17 1000
-3 21 9 12 15 17 1000
-3 22 9 12 15 18 1000
-3 23 10 13 16 19 1000
-3 24 10 14 16 19 1000
-3 25 11 14 17 20 1000
-3 26 12 15 18 21 1000
-3 27 12 15 18 21 1000
-3 28 13 16 19 22 1000
-3 29 13 16 20 23 1000
-3 30 14 17 20 23 1000
-3 31 14 18 21 24 1000
-3 32 15 18 21 25 1000
-3 33 15 19 22 25 1000
-3 34 16 19 23 26 1000
-3 35 16 20 23 27 1000
-3 36 17 21 24 27 1000
-3 37 17 21 24 28 1000
-3 38 18 22 25 29 1000
-3 39 18 22 26 29 1000
-3 40 19 23 26 30 1000
-3 41 20 24 27 31 1000
-3 42 20 24 28 31 1000
-3 43 21 25 28 32 1000
-3 44 21 25 29 33 1000
-3 45 22 26 29 33 1000
-3 46 22 26 30 34 1000
-3 47 23 27 31 35 1000
-3 48 23 28 31 35 1000
-3 49 24 28 32 36 1000
-3 50 24 29 32 37 1000
-3 51 25 29 33 37 1000
-3 52 25 30 34 38 1000
-3 53 26 31 34 39 1000
-3 54 27 31 35 39 1000
-3 55 27 32 36 40 1000
-3 56 28 32 36 41 1000
-3 57 28 33 37 41 1000
-3 58 29 34 37 42 1000
-3 59 29 34 38 43 1000
-3 60 30 35 39 43 1000
-2 1 0 0 2 2 1000
-2 2 0 1 2 3 1000
-2 3 0 1 3 4 1000
-2 4 1 2 4 5 1000
-2 5 1 2 4 6 1000
-2 6 1 3 5 6 1000
-2 7 2 3 6 7 1000
-2 8 2 4 6 8 1000
-2 9 3 5 7 8 1000
-2 10 3 5 7 9 1000
-2 11 4 6 8 10 1000
-2 12 4 6 9 11 1000
-2 13 4 7 9 11 1000
-2 14 5 7 10 12 1000
-2 15 5 8 10 13 1000
-2 16 6 8 11 13 1000
-2 17 6 9 11 14 1000
-2 18 7 9 12 15 1000
-2 19 7 10 13 15 1000
-2 20 8 11 13 16 1000
-2 21 8 11 14 17 1000
-2 22 9 12 14 17 1000
-2 23 9 12 15 18 1000
-2 24 10 13 16 19 1000
-2 25 10 13 16 19 1000
-2 26 11 14 17 20 1000
-2 27 11 14 17 21 1000
-2 28 12 15 18 21 1000
-2 29 12 15 19 22 1000
-2 30 13 16 19 22 1000
-2 31 13 17 20 23 1000
-2 32 14 17 20 24 1000
-2 33 14 18 21 24 1000
-2 34 15 18 21 25 1000
-2 35 15 19 22 26 1000
-2 36 16 19 23 26 1000
-2 37 16 20 23 27 1000
-2 38 17 20 24 28 1000
-2 39 17 21 24 28 1000
-2 40 18 22 25 29 1000
-2 41 18 22 26 29 1000
-2 42 19 23 26 30 1000
-2 43 19 23 27 31 1000
-2 44 20 24 27 31 1000
-2 45 20 24 28 32 1000
-2 46 21 25 28 33 1000
-2 47 21 25 29 33 1000
-2 48 22 26 30 34 1000
-2 49 22 27 30 35 1000
-2 50 23 27 31 35 1000
-2 51 23 28 31 36 1000
-2 52 24 28 32 36 1000
-2 53 24 29 33 37 1000
-2 54 25 29 33 38 1000
-2 55 25 30 34 38 1000
-2 56 26 30 34 39 1000
-2 57 26 31 35 40 1000
-2 58 27 32 35 40 1000
-2 59 27 32 36 41 1000
-2 60 28 33 37 41 1000
-1 1 0 0 2 2 1000
-1 2 0 1 2 3 1000
-1 3 0 1 3 4 1000
-1 4 1 2 4 5 1000
-1 5 1 2 4 5 1000
-1 6 1 3 5 6 1000
-1 7 2 3 5 7 1000
-1 8 2 4 6 8 1000
-1 9 2 4 6 8 1000
-1 10 3 5 7 9 1000
-1 11 3 5 8 10 1000
-1 12 4 6 8 10 1000
-1 13 4 6 9 11 1000
-1 14 4 7 9 12 1000
-1 15 5 7 10 12 1000
-1 16 5 8 10 13 1000
-1 17 6 8 11 13 1000
-1 18 6 9 11 14 1000
-1 19 7 9 12 15 1000
-1 20 7 10 13 15 1000
-1 21 7 10 13 16 1000
-1 22 8 11 14 17 1000
-1 23 8 11 14 17 1000
-1 24 9 12 15 18 1000
-1 25 9 12 15 18 1000
-1 26 10 13 16 19 1000
-1 27 10 13 16 20 1000
-1 28 11 14 17 20 1000
-1 29 11 14 18 21 1000
-1 30 12 15 18 21 1000
-1 31 12 16 19 22 1000
-1 32 12 16 19 23 1000
-1 33 13 17 20 23 1000
-1 34 13 17 20 24 1000
-1 35 14 18 21 25 1000
-1 36 14 18 21 25 1000
-1 37 15 19 22 26 1000
-1 38 15 19 23 26 1000
-1 39 16 20 23 27 1000
-1 40 16 20 24 28 1000
-1 41 17 21 24 28 1000
-1 42 17 21 25 29 1000
-1 43 18 22 25 29 1000
-1 44 18 22 26 30 1000
-1 45 19 23 26 31 1000
-1 46 19 23 27 31 1000
-1 47 20 24 27 32 1000
-1 48 20 24 28 32 1000
-1 49 20 25 29 33 1000
-1 50 21 25 29 33 1000
-1 51 21 26 30 34 1000
-1 52 22 26 30 35 1000
-1 53 22 27 31 35 1000
-1 54 23 27 31 36 1000
-1 55 23 28 32 36 1000
-1 56 24 29 32 37 1000
-1 57 24 29 33 38 1000
-1 58 25 30 33 38 1000
-1 59 25 30 34 39 1000
-1 60 26 31 35 39 1000
0 1 0 0 2 2 1000
0 2 0 1 2 3 1000
0 3 0 1 3 4 1000
0 4 0 2 3 5 1000
0 5 1 2 4 5 1000
0 6 1 3 4 6 1000
0 7 1 3 5 7 1000
0 8 2 3 6 7 1000
0 9 2 4 6 8 1000
0 10 2 4 7 9 1000
0 11 3 5 7 9 1000
0 12 3 5 8 10 1000
0 13 4 6 8 10 1000
0 14 4 6 9 11 1000
0 15 4 7 9 12 1000
0 16 5 7 10 12 1000
0 17 5 8 10 13 1000
0 18 6 8 11 13 1000
0 19 6 9 11 14 1000
0 20 6 9 12 15 1000
0 21 7 10 12 15 1000
0 22 7 10 13 16 1000
0 23 8 11 13 16 1000
0 24 8 11 14 17 1000
0 25 8 12 14 18 1000
0 26 9 12 15 18 1000
0 27 9 12 16 19 1000
0 28 10 13 16 19 1000
0 29 10 13 17 20 1000
0 30 11 14 17 20 1000
0 31 11 14 18 21 1000
0 32 11 15 18 22 1000
0 33 12 15 19 22 1000
0 34 12 16 19 23 1000
0 35 13 16 20 23 1000
0 36 13 17 20 24 1000
0 37 14 17 21 24 1000
0 38 14 18 21 25 1000
0 39 14 18 22 26 1000
0 40 15 19 22 26 1000
0 41 15 19 23 27 1000
0 42 16 20 23 27 1000
0 43 16 20 24 28 1000
0 44 17 21 24 28 1000
0 45 17 21 25 29 1000
0 46 17 22 25 30 1000
0 47 18 22 26 30 1000
0 48 18 23 26 31 1000
0 49 19 23 27 31 1000
0 50 19 24 27 32 1000
0 51 20 24 28 32 1000
0 52 20 25 28 33 1000
0 53 21 25 29 33 1000
0 54 21 26 29 34 1000
0 55 21 26 30 35 1000
0 56 22 27 30 35 1000
0 57 22 27 31 36 1000
0 58 23 28 31 36 1000
0 59 23 28 32 37 1000
0 60 24 29 32 37 1000
1 1 0 0 2 2 1000
1 2 0 1 2 3 1000
1 3 0 1 3 4 1000
1 4 0 1 3 4 1000
1 5 1 2 4 5 1000
1 6 1 2 4 6 1000
1 7 1 3 5 6 1000
1 8 1 3 5 7 1000
1 9 2 4 6 8 1000
1 10 2 4 6 8 1000
1 11 2 4 7 9 1000
1 12 3 5 7 9 1000
1 13 3 5 8 10 1000
1 14 3 6 8 11 1000
1 15 4 6 9 11 1000
1 16 4 7 9 12 1000
1 17 5 7 10 12 1000
1 18 5 8 10 13 1000
1 19 5 8 11 13 1000
1 20 6 8 11 14 1000
1 21 6 9 12 15 1000
1 22 6 9 12 15 1000
1 23 7 10 13 16 1000
1 24 7 10 13 16 1000
1 25 8 11 14 17 1000
1 26 8 11 14 17 1000
1 27 8 12 15 18 1000
1 28 9 12 15 18 1000
1 29 9 12 16 19 1000
1 30 10 13 16 19 1000
1 31 10 13 16 20 1000
1 32 10 14 17 21 1000
1 33 11 14 17 21 1000
1 34 11 15 18 22 1000
1 35 11 15 18 22 1000
1 36 12 16 19 23 1000
1 37 12 16 19 23 1000
1 38 13 16 20 24 1000
1 39 13 17 20 24 1000
1 40 13 17 21 25 1000
1 41 14 18 21 25 1000
1 42 14 18 22 26 1000
1 43 15 19 22 26 1000
1 44 15 19 23 27 1000
1 45 15 20 23 27 1000
1 46 16 20 24 28 1000
1 47 16 21 24 28 1000
1 48 17 21 25 29 1000
1 49 17 21 25 30 1000
1 50 18 22 26 30 1000
1 51 18 22 26 31 1000
1 52 18 23 27 31 1000
1 53 19 23 27 32 1000
1 54 19 24 28 32 1000
1 55 20 24 28 33 1000
1 56 20 25 28 33 1000
1 57 20 25 29 34 1000
1 58 21 26 29 34 1000
1 59 21 26 30 35 1000
1 60 22 26 30 35 1000
2 1 0 0 2 2 1000
2 2 0 1 2 3 1000
2 3 0 1 3 4 1000
2 4 0 1 3 4 1000
2 5 0 2 4 5 1000
2 6 1 2 4 6 1000
2 7 1 2 5 6 1000
2 8 1 3 5 7 1000
2 9 2 3 5 7 1000
2 10 2 4 6 8 1000
2 11 2 4 6 8 1000
2 12 2 4 7 9 1000
2 13 3 5 7 10 1000
2 14 3 5 8 10 1000
2 15 3 6 8 11 1000
2 16 4 6 9 11 1000
2 17 4 7 9 12 1000
2 18 4 7 10 12 1000
2 19 5 7 10 13 1000
2 20 5 8 10 13 1000
2 21 5 8 11 14 1000
2 22 6 9 11 14 1000
2 23 6 9 12 15 1000
2 24 6 9 12 15 1000
2 25 7 10 13 16 1000
2 26 7 10 13 16 1000
2 27 7 11 14 17 1000
2 28 8 11 14 17 1000
2 29 8 11 15 18 1000
2 30 9 12 15 18 1000
2 31 9 12 15 19 1000
2 32 9 13 16 19 1000
2 33 10 13 16 20 1000
2 34 10 14 17 20 1000
2 35 10 14 17 21 1000
2 36 11 14 18 21 1000
2 37 11 15 18 22 1000
2 38 11 15 19 22 1000
2 39 12 16 19 23 1000
2 40 12 16 19 23 1000
2 41 13 16 20 24 1000
2 42 13 17 20 24 1000
2 43 13 17 21 25 1000
2 44 14 18 21 25 1000
2 45 14 18 22 26 1000
2 46 14 19 22 26 1000
2 47 15 19 23 27 1000
2 48 15 19 23 27 1000
2 49 15 20 23 28 1000
2 50 16 20 24 28 1000
2 51 16 21 24 29 1000
2 52 17 21 25 29 1000
2 53 17 21 25 30 1000
2 54 17 22 26 30 1000
2 55 18 22 26 31 1000
2 56 18 23 27 31 1000
2 57 18 23 27 32 1000
2 58 19 24 27 32 1000
2 59 19 24 28 33 1000
2 60 20 24 28 33 1000
3 1 0 0 2 2 1000
3 2 0 0 2 3 1000
3 3 0 1 2 4 1000
3 4 0 1 3 4 1000
3 5 0 2 3 5 1000
3 6 1 2 4 5 1000
3 7 1 2 4 6 1000
3 8 1 3 5 6 1000
3 9 1 3 5 7 1000
3 10 2 3 6 8 1000
3 11 2 4 6 8 1000
3 12 2 4 6 9 1000
3 13 2 4 7 9 1000
3 14 3 5 7 10 1000
3 15 3 5 8 10 1000
3 16 3 6 8 11 1000
3 17 4 6 9 11 1000
3 18 4 6 9 12 1000
3 19 4 7 9 12 1000
3 20 4 7 10 13 1000
3 21 5 7 10 13 1000
3 22 5 8 11 14 1000
3 23 5 8 11 14 1000
3 24 6 9 11 15 1000
3 25 6 9 12 15 1000
3 26 6 9 12 15 1000
3 27 7 10 13 16 1000
3 28 7 10 13 16 1000
3 29 7 10 14 17 1000
3 30 8 11 14 17 1000
3 31 8 11 14 18 1000
3 32 8 12 15 18 1000
3 33 9 12 15 19 1000
3 34 9 12 16 19 1000
3 35 9 13 16 20 1000
3 36 10 13 16 20 1000
3 37 10 14 17 21 1000
3 38 10 14 17 21 1000
3 39 11 14 18 22 1000
3 40 11 15 18 22 1000
3 41 11 15 18 22 1000
3 42 12 15 19 23 1000
3 43 12 16 19 23 1000
3 44 12 16 20 24 1000
3 45 13 17 20 24 1000
3 46 13 17 21 25 1000
3 47 13 17 21 25 1000
3 48 14 18 21 26 1000
3 49 14 18 22 26 1000
3 50 14 19 22 27 1000
3 51 15 19 23 27 1000
3 52 15 19 23 28 1000
3 53 15 20 23 28 1000
3 54 16 20 24 28 1000
3 55 16 20 24 29 1000
3 56 16 21 25 29 1000
3 57 17 21 25 30 1000
3 58 17 22 25 30 1000
3 59 17 22 26 31 1000
3 60 18 22 26 31 1000
4 1 0 0 2 2 1000
4 2 0 0 2 3 1000
4 3 0 1 2 3 1000
4 4 0 1 3 4 1000
4 5 0 1 3 5 1000
4 6 0 2 4 5 1000
4 7 1 2 4 6 1000
4 8 1 2 4 6 1000
4 9 1 3 5 7 1000
4 10 1 3 5 7 1000
4 11 2 3 6 8 1000
4 12 2 4 6 8 1000
4 13 2 4 6 9 1000
4 14 2 4 7 9 1000
4 15 3 5 7 10 1000
4 16 3 5 8 10 1000
4 17 3 5 8 11 1000
4 18 3 6 8 11 1000
4 19 4 6 9 11 1000
4 20 4 6 9 12 1000
4 21 4 7 9 12 1000
4 22 4 7 10 13 1000
4 23 5 7 10 13 1000
4 24 5 8 11 14 1000
4 25 5 8 11 14 1000
4 26 6 8 11 15 1000
4 27 6 9 12 15 1000
4 28 6 9 12 15 1000
4 29 6 10 13 16 1000
4 30 7 10 13 16 1000
4 31 7 10 13 17 1000
4 32 7 11 14 17 1000
4 33 8 11 14 18 1000
4 34 8 11 14 18 1000
4 35 8 12 15 18 1000
4 36 8 12 15 19 1000
4 37 9 12 16 19 1000
4 38 9 13 16 20 1000
4 39 9 13 16 20 1000
4 40 10 13 17 21 1000
4 41 10 14 17 21 1000
4 42 10 14 17 21 1000
4 43 11 14 18 22 1000
4 44 11 15 18 22 1000
4 45 11 15 19 23 1000
4 46 11 15 19 23 1000
4 47 12 16 19 24 1000
4 48 12 16 20 24 1000
4 49 12 17 20 24 1000
4 50 13 17 20 25 1000
4 51 13 17 21 25 1000
4 52 13 18 21 26 1000
4 53 14 18 22 26 1000
4 54 14 18 22 27 1000
4 55 14 19 22 27 1000
4 56 15 19 23 27 1000
4 57 15 19 23 28 1000
4 58 15 20 24 28 1000
4 59 16 20 24 29 1000
4 60 16 20 24 29 1000
5 1 0 0 1 2 1000
5 2 0 0 2 3 1000
5 3 0 1 2 3 1000
5 4 0 1 3 4 1000
5 5 0 1 3 4 1000
5 6 0 1 3 5 1000
5 7 0 2 4 5 1000
5 8 1 2 4 6 1000
5 9 1 2 4 6 1000
5 10 1 3 5 7 1000
5 11 1 3 5 7 1000
5 12 1 3 6 8 1000
5 13 2 4 6 8 1000
5 14 2 4 6 9 1000
5 15 2 4 7 9 1000
5 16 2 5 7 9 1000
5 17 3 5 7 10 1000
5 18 3 5 8 10 1000
5 19 3 5 8 11 1000
5 20 3 6 8 11 1000
5 21 4 6 9 12 1000
5 22 4 6 9 12 1000
5 23 4 7 9 12 1000
5 24 4 7 10 13 1000
5 25 5 7 10 13 1000
5 26 5 8 11 14 1000
5 27 5 8 11 14 1000
5 28 5 8 11 14 1000
5 29 6 9 12 15 1000
5 30 6 9 12 15 1000
5 31 6 9 12 16 1000
5 32 6 10 13 16 1000
5 33 7 10 13 17 1000
5 34 7 10 13 17 1000
5 35 7 11 14 17 1000
5 36 7 11 14 18 1000
5 37 8 11 14 18 1000
5 38 8 11 15 18 1000
5 39 8 12 15 19 1000
5 40 9 12 15 19 1000
5 41 9 12 16 20 1000
5 42 9 13 16 20 1000
5 43 9 13 16 20 1000
5 44 10 13 17 21 1000
5 45 10 14 17 21 1000
5 46 10 14 17 22 1000
5 47 10 14 18 22 1000
5 48 11 15 18 22 1000
5 49 11 15 19 23 1000
5 50 11 15 19 23 1000
5 51 12 16 19 24 1000
5 52 12 16 20 24 1000
5 53 12 16 20 24 1000
5 54 12 17 20 25 1000
5 55 13 17 21 25 1000
5 56 13 17 21 26 1000
5 57 13 18 21 26 1000
5 58 14 18 22 26 1000
5 59 14 18 22 27 1000
5 60 14 19 22 27 1000
6 1 0 0 1 2 1000
6 2 0 0 2 3 1000
6 3 0 1 2 3 1000
6 4 0 1 3 4 1000
6 5 0 1 3 4 1000
6 6 0 1 3 5 1000
6 7 0 2 4 5 1000
6 8 0 2 4 6 1000
6 9 1 2 4 6 1000
6 10 1 2 5 6 1000
6 11 1 3 5 7 1000
6 12 1 3 5 7 1000
6 13 1 3 6 8 1000
6 14 2 4 6 8 1000
6 15 2 4 6 9 1000
6 16 2 4 6 9 1000
6 17 2 4 7 9 1000
6 18 2 5 7 10 1000
6 19 3 5 7 10 1000
6 20 3 5 8 11 1000
6 21 3 5 8 11 1000
6 22 3 6 8 11 1000
6 23 3 6 9 12 1000
6 24 4 6 9 12 1000
6 25 4 7 9 12 1000
6 26 4 7 10 13 1000
6 27 4 7 10 13 1000
6 28 5 7 10 14 1000
6 29 5 8 11 14 1000
6 30 5 8 11 14 1000
6 31 5 8 11 15 1000
6 32 6 9 12 15 1000
6 33 6 9 12 15 1000
6 34 6 9 12 16 1000
6 35 6 9 13 16 1000
6 36 6 10 13 17 1000
6 37 7 10 13 17 1000
6 38 7 10 14 17 1000
6 39 7 11 14 18 1000
6 40 7 11 14 18 1000
6 41 8 11 14 18 1000
6 42 8 11 15 19 1000
6 43 8 12 15 19 1000
6 44 8 12 15 19 1000
6 45 9 12 16 20 1000
6 46 9 13 16 20 1000
6 47 9 13 16 20 1000
6 48 9 13 17 21 1000
6 49 10 14 17 21 1000
6 50 10 14 17 22 1000
6 51 10 14 18 22 1000
6 52 10 14 18 22 1000
6 53 11 15 18 23 1000
6 54 11 15 19 23 1000
6 55 11 15 19 23 1000
6 56 11 16 19 24 1000
6 57 12 16 20 24 1000
6 58 12 16 20 24 1000
6 59 12 16 20 25 1000
6 60 12 17 20 25 1000