Puzzles
7 December
In September, my puzzle appeared as
Alex Bellos's Monday Puzzle. The puzzle asked what the highest rugby score was which can only be made with one combination of kicks, tries and converted tries.
What is the highest rugby score which can be made with 101 different combinations of kicks, tries and converted tries?
Shooting hoops
You spend an afternoon practising throwing a basketball through a hoop.
One hour into the afternoon, you have scored less than 75% of your shots. At the end of the afternoon, you have score more than 75% of your shots.
Is there a point in the afternoon when you had scored exactly 75% of your shots?
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If there is not a point when you had scored exactly 75%, then there must be a shot which you score which took your average over 75%. In other words, there must be an $a$ and a $b$ so that:
$$\frac{a}{b}<\frac{3}{4}<\frac{a+1}{b+1}$$
Rearranging these inequalities gives:
$$4a<3b$$
$$3b+3<4a+4$$
Taking 3 from the bottom inequality gives:
$$4a<3b$$
$$3b<4a+1$$
Combining the inequalities gives:
$$4a<3b<4a+1$$
Therefore \(3b\) must be an integer which is between \(4a\) and \(4a+1\). But \(4a\) and \(4a+1\) are consecutive numbers so there is no other integer between them.
This means that this situation is impossible and there must have been a time when you had scored exactly 75% of your shots.
Extension
You spend another afternoon practising throwing a basketball through a hoop.
One hour into the afternoon, you have scored more than 75% of your shots. At the end of the afternoon, you have score less than 75% of your shots.
Is there a point in the afternoon when you had scored exactly 75% of your shots?
Rugby scores
In a rugby (union) match, 3 point are scored for a kick, 5 for a try and 7 for a converted try. This scoring system means that some total scores can be achieved in different combinations, while others can be achieved in only one way.
For example, 14 can be scored in two ways (three kicks and a try; or two converted tries), while 8 can only be achieved in one way (try and a kick).
What is the highest score which can only be made in one way?
What is the highest score which can be made in two ways?
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11 is the highest score which can only be made in one way (try and two kicks).
Each of 12, 13 and 14 can be made in two ways:
- 12 - try and converted try; or four kicks
- 13 - converted try and two kicks; or two tries and a kick
- 14 - three kicks and a try; or two converted tries
Now any score higher than 14 can be made by adding kicks to one of these three. This gives at least two ways of making every score higher than this.
By a similar argument, it can be shown that 16 is the highest score which can only be made in two ways.
For the highest score which can be made in \(n\) ways, see the OEIS sequence
A261155.
Extension
Which is the highest rugby score such that if the number of tries scored is known, you can always tell how many of these tries were converted?
Tennis
What is the minimum number of times a player has to hit the ball in a set of tennis and win a standard set (the set is not ended by injury, disqualification, etc.)?
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A set can be won by hitting the ball 12 times: three service games can be won with four aces, the opponent loses their service games with four double faults.
