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A Hamlet timeline – chronicle of events

      Setting the Timeline – considerations

Claudius – planning my foul murder

King Hamlet's funeral – where was Hamlet?

Gertrude & Claudius – adultery or not?

Horatio – Hamlet's friend?

Horatio – is he passion's slave?

Polonius – the evil that men do

Ophelia's love? – did she love Hamlet?

Ophelia closetted – Polonius on love

      O help xxx ....... – Olivier's version

Ophelia's change – is Hamlet suspicious?

Hamlet feigns madness – protective "cover"

Is Hamlet mad? – Polonius's opinion

Hamlet kills Polonius – stabs the "Voice"

Laertes on Ophelia – madness & death

Ophelia's death – a recipe

Hamlet's age – digging up the past

Yorick – something rotting in Denmark

Betting on Hamlet – the fencing match

Hamlet's fencing skill – better than Laertes

Betting on Hamlet

The fencing match

Claudius: We'll make a solemn wager on your cunnings:
This is part of Claudius's scheme as a cover for Hamlet's murder. If things go according to plan, Hamlet will appear to have been killed accidently by Laertes's unbated foil. By wagering on the contest, it is given the air of plausibility and normality. Although it is unlikely anyone would suspect Hamlet had been killed by foul play, such thoughts would be allayed when it was realised there was a solemn wager on the contest. It is a brilliant idea by Claudius, but having decided to do it, and to preclude the least suspicion, it requires that the wagering makes sense when it comes to assessing the odds.
Osric: The king, sir, hath laid, that in a dozen passes
between yourself and him (Laertes), he shall not exceed you
three hits: he hath laid on twelve for nine;

Claudius: Cousin Hamlet, you know the wager?
Hamlet: Very well, my lord
Your grace hath laid the odds o' the weaker side.
Claudius: I do not fear it; I have seen you both:
But since he is better'd, we have therefore odds.
Laertes is nominally the better fencer but Claudius wagers on Hamlet. He is not concerned, because this is a handicap event in that Laertes has to score 8 hits whereas Hamlet need only score 5. Once either attains his required number of hits (ie passes) the contest comes to an end. Hamlet is considered to have less chance of scoring 5 than Laertes has of reaching 8, and so Claudius has been compensated by being given odds of 12 to 9. The score-line may require as many as a dozen passes: Assume the score line after 11 passes is Laertes 7 and Hamlet 4. One more play will determine the outcome - either Laertes 8 or Hamlet 5.

Despite the apparent simplicity of the wagering, not one of the many attempts in decades, if not centuries, has managed to explain it in a meaningful way.

Achieving expectation

Consider the first part of the challenge:
Osric: The king, sir, hath laid, that in a dozen passes
between yourself and him, he shall not exceed you three hits;
A thing that has failed to be noted by any who have tried to solve the puzzle of the wagering is that in any contest of an even number of points, eg 12, the final score-line can only be an even difference. So, in this fencing match, Laertes cannot exceed Hamlet by three hits, it has to be an even number. Why then is it stated as three hits? This is not calculated by mathematicians, statisticians or logicians, but by Shakespearean characters who don't necessarily get things spot on. Perhaps it is possible to find out how they arrived at their figures.

The terms of the challenge are undoubtedly related to their expectations over 12 games. The ratio of the expectations would not change should the number of passes be doubled. If the contest had been over 24 passes the challenge would have been:
Osric: The king, sir, hath laid, that in a two dozen passes
between yourself and him, he shall not exceed you six hits;
If 24 passes were to be played, there are a number of possible score-lines. Whatever score-line is the final result determines the expectations. These are listed in the table (24 passes).
24 passes
Laertes winsHamlet winsexceedsHitsBet
12
12
  0
24
Even money
13
11
  2
24
Even money
14
10
  4
24
Even money
15
  9
  6
24
Even money
16
  8
  8
24
Even money
17
  7
10
24
Even money
18
  6
12
24
Even money
19
  5
14
24
Even money
20
  4
16
24
Even money
21
  3
18
24
Even money
22
  2
20
24
Even money
23
  1
22
24
Even money
24
  0
24
24
Even money
12 passes
Laertes winsHamlet winsexceedsHits
Bet
  6   
6   
  0
12
Even money
  6.5
5.5
  1
11
Odds
  7   
5   
  2
12
Even money
  7.5
4.5
  3
11
Odds
  8   
4   
  4
12
Even money
  8.5
3.5
  5
11
Odds
  9   
3   
  6
12
Even money
  9.5
2.5
  7
11
Odds
10   
2   
  8
12
Even money
10.5
1.5
  9
11
Odds
11   
1   
10
12
Even money
11.5
0.5
11
11
Odds
12   
0   
12
12
Even money

In a 24 passes contest, both players are equally likely to reach their expectations. Equality of expectation is the substance of all bets at "even money", ie, 1 for 1.
Claudius bets that in 12 passes Laertes will not exceed Hamlet by 3 hits.
Claudius, who appears to understand odds, might say "Hang on a minute. Although halving the passes and hits looks ok, in fact, it raises a problem. If you play 12 passes it is impossible to exceed Hamlet three hits. An even number of passes can only ever be exceeded by an even number of hits. The two tables, respectively, show all possible score-lines for a match of 24 passes and likewise for a match of 12 passes."

"Of course," continues Claudius, "in any contest, regardless of what stage or moment in time it has reached, the expectations still hold. Therefore, when 12 passes have elapsed, in an exact mathematical sense, the expectations are for Laertes to have scored 7·5 hits and Hamlet 4·5 hits. The 12th hit to actually conclude the contest would still need to be made by one of the contestants."

As was pointed out above, the odds to achieve expectation, in this case 7·5 to 4·5, is exactly even money, but once that situation has been reached, both players are still in need of 0·5 of a point. What odds apply to attain the 12th hit? The Laertes to Hamlet scoring rates are 15 to 9 and therefore their expectations are 0·5 to 0·3 respectively, so it can seen that Laertes is expected to reach his winning score before Hamlet gets his, i.e., 8 hits whilst Hamlet only gets to 4·8 hits. The disadvantage of this lesser chance needs to be compensated for by an adjustment of the odds.

Claudius tells us that he has odds of 12 for 9. But ought not those odds be 15 for 9 to be proportional to their chances of completing the 12th game? Did Claudius make a mistake?

Here is the output of a computer program that simulated 10 cycles of 1000 fencing matches.


Claudius is a shrewd gambler and understands the value of odds. What makes this a particularly beautiful bet is that it is not apparent that he has the betting advantage. If anything, the advantage seems to lie with the other bettor, not Claudius. Firstly, that it looks as though Claudius has taken under the odds, and secondly, that he has wagered against Laertes, a swordsman known to be highly-skilled.

In the actual fencing match, despite the acknowledged skill of Laertes, Hamlet proves to be far better. How come that suddenly Hamlet has become an expert swordsman?
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