Sports / Maths Puzzle(s)...

Matt Bayfield · Post by **Matt Bayfield** » Tue Dec 29, 2009 12:34 pm

About a year ago, I set a few mathematical puzzles based on statistics from sports (and games). I hope that one or two of them might appeal to folks round here, so I'll post them here over the next few weeks. Some very, very basic sporting knowledge is required for some of the puzzles (e.g. to solve the first puzzle, you need to know that in a game of darts, each player throws three darts per turn).

The first puzzle was set when I was watching the Grand Slam of Darts in November 2008. Post your answers in the thread, although if you want to avoid spoiling the puzzle for other people, feel free to post your answer in yellow or some other colour which doesn't show up readily until you select the text.

PUZZLE ONE: DARTS

In darts, a player’s “three-dart average” over a match is defined as the total points a player scores during the match, divided by the total number of darts he throws, then multiplied by three.

In the Grand Slam of Darts, the 2nd round matches are played as the first to 10 standard legs of 501, with one player throwing first in all the even-numbered legs and the other player throwing first in all the odd-numbered legs (i.e. they throw first alternately). As in normal professional darts matches, each leg must be finished on a double or the inner bull (but there is no requirement to start on a double, as played in some tournaments).

What is the maximum (theoretical) three-dart average which a player could achieve in the 2nd round of the Grand Slam of Darts?

Assume that the match is played to a normal conclusion, i.e. no player retires early, or is disqualified, or anything daft like that.

David Roe · Post by **David Roe** » Tue Dec 29, 2009 1:13 pm

Is it 10 x 9 dart finishes, 9 x 360 scores as your opponent does 9 x 9 dart finishes, overall total 8250 points, average 171.87?

[Test for colour, first time I've done this.] OK.

Matt Bayfield · Post by **Matt Bayfield** » Tue Dec 29, 2009 2:07 pm

David: you've given the answer that many people gave when I originally set this puzzle - and I'm afraid it's not right. I won't give any further clues at this stage, but try again if you like...

Hugh Binnie · Post by **Hugh Binnie** » Tue Dec 29, 2009 2:31 pm

The trick is to notice that you can have a higher average if you lose a frame than if you win. Bearing this in mind, hopefully the average is maximized when the player goes second and makes 9 9-dart checkouts but loses to 10 9-dart checkouts, having scored 360 in each loss, which gives an average of 172.53.

Matt Bayfield · Post by **Matt Bayfield** » Tue Dec 29, 2009 2:53 pm

Hugh: that's spot on. Well worked out.

Andrew Hulme · Post by **Andrew Hulme** » Tue Dec 29, 2009 6:14 pm

not that this matters for the purposes of this puzzle i guess, and if i'm wrong i might get lynched by the dart fans on here... but im fairly sure ur defn of a players 3-dart average isnt correct. shouldnt it be: total points scored/no of visits to board ? thats certainly what I've always understood it to mean when I've seen in on TV anyway?

Also, an easy one to add to this: What's the lowest score a player can achieve with their first dart and still achieve a 9-dart finish?

Matthew Green · Post by **Matthew Green** » Tue Dec 29, 2009 6:17 pm

Double 17?

Kirk Bevins · Post by **Kirk Bevins** » Tue Dec 29, 2009 6:18 pm

Andrew Hulme wrote:not that this matters for the purposes of this puzzle i guess, and if i'm wrong i might get lynched by the dart fans on here... but im fairly sure ur defn of a players 3-dart average isnt correct. shouldnt it be: total points scored/no of visits to board ? thats certainly what I've always understood it to mean when I've seen in on TV anyway?

Also, an easy one to add to this: What's the lowest score a player can achieve with their first dart and still achieve a 9-dart finish?

I'd guess 151 (151, 180, 170). Anyway it's not number of visits to the board. For a start if you want double 16 and hit it with the first dart or second dart, your average will be different. With your method it will be the same as it's just one visit to the board.

Edit: I didn't read your question right. I think double 17 is the correct answer too.

Matt Bayfield · Post by **Matt Bayfield** » Tue Dec 29, 2009 7:42 pm

Andrew Hulme wrote:not that this matters for the purposes of this puzzle i guess, and if i'm wrong i might get lynched by the dart fans on here... but im fairly sure ur defn of a players 3-dart average isnt correct. shouldnt it be: total points scored/no of visits to board ? thats certainly what I've always understood it to mean when I've seen in on TV anyway?

Andrew: No lynching necessary, but the definition I've given above, is what's used pretty much universally (including on all tv coverage) for 3-dart average. So if a player throws a 15-dart leg, his average is 3 x 501/15 = 100.2, and for a 14-dart leg, his average is 3 x 501/14. If you watch Sky coverage (and poss also BBC or ITV4), where the average is sometimes displayed and updated in realtime as a player throws each of his three darts, you'll see that this must be the case (because the 3-dart average will go either up or down depending on whether the score of each dart is greater or less than one-third of the current three-dart average).

The only slight caveat is that if a player busts a shot on his first or second dart of a throw, then I'm fairly sure that counts as all three darts thrown for the purposes of calculating averages.

[Edit: correction - "3 x" inserted in two places in my first paragraph.]

Kirk Bevins · Post by **Kirk Bevins** » Tue Dec 29, 2009 8:54 pm

Matt Bayfield wrote:
The only slight caveat is that if a player busts a shot on his first or second dart of a throw, then I'm fairly sure that counts as all three darts thrown for the purposes of calculating averages.

Correct.

David Roe · Post by **David Roe** » Tue Dec 29, 2009 10:59 pm

I thought I'd just squeezed it up to 172.63, but that was relying on scoring 500 in a losing frame. But I think 500 counts as bust, doesn't it? So it won't work - the 4 points lost brings it down to 172.48.

Matt Bayfield · Post by **Matt Bayfield** » Wed Dec 30, 2009 12:33 pm

David: You're correct about a player busting his score if he scores 500 points in a leg. Still, I'm not quite sure how you're working out those numbers - as it would take you 9 darts to reach 500 (or 499) in a losing leg, which would be a worse average than scoring 360 over 6 darts. Obviously you could score 480 after 8 darts, but you would still have to throw the ninth, as darts are thrown in sets of three, and you could only score a maximum of 19 with that ninth dart in a losing leg, otherwise you would win the leg or bust your score.

Incidentally, if anyone thinks I've got an answer wrong at any point in this series of puzzles, please do let me know (although I would hope there are no mistakes since these puzzles were previously tested on people last year).

Anyhow:

ANSWER TO PUZZLE ONE

Although relatively straightforward to calculate, the solution to Puzzle One is perhaps counter-intuitive to non-darts players, as you might ordinarily expect the winning player to have the higher average. However, those who follow darts more closely will know that it’s the finishing that brings your average down (because you have to check out with whatever score is left, rather than hit the maximum possible with each dart). This makes it more understandable that a player who finishes fewer times (i.e. loses 10-9) will have a higher average than a player who wins 10-9, when all legs are nine-dart finishes.

Answer: 172.53 (first correct solution: Hugh)

Today's sport/maths teaser is another fairly straightforward one, this time on the subject of English Premier League football. (For those who don’t feel particularly challenged by these puzzles, there will be some trickier ones coming along soon, including an absolutely fiendish one on the sport of sailing...)

PUZZLE TWO: FOOTBALL

In a Premier League season with 20 teams, each team plays every other team twice (once at home and once away). The teams finishing in the bottom 3 positions at the end of the season are then relegated to the Championship. During the season, teams gain 3 points for a win and 1 for a draw. What is the maximum number of points a team can score in a Premier League season and still be relegated?

In this calculation, ignore the effect of any points adjustments or enforced relegations due to financial irregularities, ineligible players, etc.

Dinos Sfyris · Post by **Dinos Sfyris** » Wed Dec 30, 2009 12:55 pm

Love the first puzzle. I had the exact same thought as David Roe. I assumed to maximise the average they would win the 2nd round 10-9. Didn't realise the losing player could in fact have the higher average. Will get cracking on the football one now.

Dinos Sfyris · Post by **Dinos Sfyris** » Wed Dec 30, 2009 1:01 pm

What happens if teams finishing 17th and 18th are on equal points?

David Roe · Post by **David Roe** » Wed Dec 30, 2009 1:09 pm

The darts one: if you lose the match 10-1, you can score 360 in losing the 6 odd numbered games, 500 (well, you can't, but you get what I mean) in 9 darts in 4 of the losses, and 501 9 dart finish in your only win. This gives 4661 points in 81 darts, ave. 172.63.
When your max in the four losses is 499, the 4 points you lose make the average 172.48.

Premier League football: 2 teams lose every game, so the relegated side gets 12 points off them. The other 18 beat each other and lose to each other once each, hence 51 points in 34 games each. So all 18 top teams get 63 points, including one of the relegated teams; the other two teams share up to 6 between them.

Dinos Sfyris · Post by **Dinos Sfyris** » Wed Dec 30, 2009 1:16 pm

Ok after some quick workings out here's my first answer:

Obviously each team plays 38 matches in total, 2 against each of the other 19 teams. To maximise points total for the team finishing 18th the bottom 2 teams should win no games at all except for the 2 games between themselves for which the outcome doesn't matter. The remaining 18 teams should all be as close as possible so if they all win one and lose one against each team except for the bottom 2 teams which gives them 4 extra victories then they should have 21 wins, 17 losses each (63 points) and the team placed 18th with the worst goal difference should be relegated.

Gavin Chipper · Post by **Gavin Chipper** » Wed Dec 30, 2009 5:29 pm

Matt Bayfield wrote:(For those who don’t feel particularly challenged by these puzzles, there will be some trickier ones coming along soon, including an absolutely fiendish one on the sport of sailing...)

Looking forward to it. I like the ones where no-one can get them straight away so when you turn up it won't necessarily have been solved already, and people work off each other's partial solutions until someone finally drives it home.

Matt Bayfield · Post by **Matt Bayfield** » Wed Dec 30, 2009 5:38 pm

David: re: the darts puzzle, I have to admit I hadn't even thought of the 10-1 loss scoreline, so thanks for clarifying. I'm seriously impressed with your lateral thinking, but I'm glad that my answer still holds up! (Although I'd also have been happy to have been proven wrong, if I had been wrong. Part of the fun of setting these kinds of puzzle.)

Re: the football puzzle, both David and Dinos are correct.

Gavin: the sailing puzzle (coming soon) is particularly interesting, as although I'm fairly close to a solution, it'll need someone more mathematical than me, or seriously good with a computer, to prove it's the correct solution!

David Roe · Post by **David Roe** » Wed Dec 30, 2009 11:06 pm

Bring them on! Sport and numbers, the perfect combination. One a day for the next year would seem about right.

Matt Bayfield · Post by **Matt Bayfield** » Thu Dec 31, 2009 12:47 pm

David: a whole year's puzzles might be pushing it... but at the current rate of one a day, I've probably got enough to last til the end of the first week of January.

ANSWER TO PUZZLE TWO

Pretty much as stated above, my logic was as follows:

Maximum points scored in total in the season so no draws.
Each team plays 38 games (19 home, 19 away).
Bottom 2 sides each lose home and away to the top 18 teams.
Leaving 34 other games for the top 18 teams (17 home, 17 away).
Each top 18 team wins all their home games.
Points scored by a top 18 team is therefore: 17 x 3 = 51 points (against other top 18 teams) + (4 x 3) = 12 pts against bottom 2, total 63 points.
All top 18 teams finish level on 63 pts, the team relegated in 18th place will be the one with the worst goal difference.

Answer: 63 (first correct solution: David)

Now on to today's puzzle, which is trickier than either of the first two. Knowledge of the scoring system in tennis is required.

PUZZLE THREE: TENNIS

In a gentlemen's singles match at Wimbledon, what is the minimum percentage of the total points played that a player can win, and still win the match? Give the answer to 2 decimal places.

Assumptions:
(1) the match is played to normal completion (i.e. no player retires hurt, gets disqualified, etc.)
(2) no penalty points are awarded at any stage for code violations (e.g. swearing, racket abuse).
(3) the match is played under current scoring rules at Wimbledon, i.e. best of five sets, with tie-breaks played in the first four sets only should the score reach 6 games all.

[Edit: correction - "Grand Slam" corrected to "Wimbledon" in Assumption (3) as per Charlie's comment below.]

Oliver Garner · Post by **Oliver Garner** » Thu Dec 31, 2009 12:57 pm

Matt Bayfield wrote:
PUZZLE THREE: TENNIS

In a gentlemen's singles match at Wimbledon, what is the minimum percentage of the total points played that a player can win, and still win the match? Give the answer to 2 decimal places.

Assumptions:
(1) the match is played to normal completion (i.e. no player retires hurt, gets disqualified, etc.)
(2) no penalty points are awarded at any stage for code violations (e.g. swearing, racket abuse).
(3) the match is played under current scoring rules in Grand Slam tournaments, i.e. best of five sets, with tie-breaks played in the first four sets only should the score reach 6 games all.

Answer : 0.00% - If one player wins all his games to love and the other player wins all his games with a sufficiently high amount of deuces, the percentage of points won would be below 0.005% and therefore be 0.00%

Matt Bayfield · Post by **Matt Bayfield** » Thu Dec 31, 2009 1:06 pm

Oliver: it's not that simple, I'm afraid. I won't say any more for now, to avoid giving clues.

Charlie Reams · Post by **Charlie Reams** » Thu Dec 31, 2009 1:08 pm

Matt Bayfield wrote: (3) the match is played under current scoring rules in Grand Slam tournaments, i.e. best of five sets, with tie-breaks played in the first four sets only should the score reach 6 games all.

FWIW only Wimbledon works this way.

Matt Bayfield · Post by **Matt Bayfield** » Thu Dec 31, 2009 1:10 pm

Cheers Charlie. I shall edit the question, changing "Grand Slam" to "Wimbledon".

Charlie Reams · Post by **Charlie Reams** » Thu Dec 31, 2009 1:16 pm

Matt Bayfield wrote:Cheers Charlie. I shall edit the question, changing "Grand Slam" to "Wimbledon".

I only posted that to compensate for my inferiority at being unable to answer the question

Clive Brooker · Post by **Clive Brooker** » Thu Dec 31, 2009 1:53 pm

Risking embarrassment (tennis):

Two sets are lost to 6 love games - 0-48 (2 of 0-24) in terms of points
Four games in the three sets won are lost to love - again 0-48 in points (3 of 0-16)
Six games in three sets are won to 30 (4-2) - 72-36 in points

The total points tally is 72-132, the winner's percentage being 35.3

As far as I can see, extending any games into deuce or extending sets beyoned 6-0 or 6-4 can only cause the points tallies to converge.

JimBentley · Post by **JimBentley** » Thu Dec 31, 2009 2:12 pm

OK, I don't think this is quite right, but hey:

Player 1 goes on a rampage for the first two sets, winning every point (every game goes 15-0, 30-0, 40-0, game), so by the end of the second set he leads 6-0 6-0, having won 48 points and lost none.

Player 2 then has a comeback of sorts, in which he wins 6 games each 4 points to 2 (15-0, 15-15, 30-15, 30-30, 40-30, game) but loses 4 games as in the first two sets (0-15, 0-30, 0-40, game), so from the third set Player 1 has won 28 points from 52 and Player 2 has won 24 points from 52.

This is then repeated in the the fourth and fifth sets, so Player 2 wins the match 0-6, 0-6, 6-4, 6-4, 6-4. Player 1 has won 48+28+28+28 = 132 points and Player 2 has won 0+24+24+24 = 72 points, giving Player 2 72/204 points, or 35.29%.

Excellent puzzles these Matt!

Edit: oh bollocks Clive beat me to it, well done sir.

Matt Bayfield · Post by **Matt Bayfield** » Thu Dec 31, 2009 2:35 pm

Clive & Jim (who have both given the same response): very close, and you're thinking along the right lines... but wrong! (Sorry, that sounds like I'm enjoying this...

)

Another go? Or anyone else?

Clive Brooker · Post by **Clive Brooker** » Thu Dec 31, 2009 3:19 pm

I should have checked more carefully first time.

I think I can get it down to 35.25%

By extending the first two sets won to a tie-break, the winner's minimum point-count is increased by 7 in each set, the loser's corresponding maximum by 13 in each set.

The total points are now 86 - 158

Matt Bayfield · Post by **Matt Bayfield** » Thu Dec 31, 2009 3:33 pm

Clive: good work fella - that's the correct answer.

David Roe · Post by **David Roe** » Thu Dec 31, 2009 4:03 pm

Without checking anyone else's answer, my first go is:
Lose 2 sets by 0-6, scoring 0 points to 24. Win 2 sets on tie breaks, scoring 31 points and conceding 41 - 4 in each lost game, 2 in each won game, 5 in the tie break. Win the last set 6-4, scoring 24 points in the 6 wins, conceding 28 - 2 in each won game, 4 in each loss.

Total points 86 won, 158 lost, 35.25%.

[Edit] Hooray.

Gavin Chipper · Post by **Gavin Chipper** » Thu Dec 31, 2009 4:56 pm

Yeah, I did it the same as Clive and David so I'm claiming my share of the jackpot.

David Roe · Post by **David Roe** » Fri Jan 01, 2010 10:16 am

I'm willing to share my share, Gavin.

Matt Bayfield · Post by **Matt Bayfield** » Mon Jan 04, 2010 4:58 pm

ANSWER TO PUZZLE THREE

This is the long-winded way I described my answer when i originally set this puzzle just over a year ago:

This problem is complicated because of the unusual scoring system in tennis. To solve it, you need to consider points won in a game, then each set, then the match - in that order.

First lets consider a game.
---------------------------

If a player, Billy Muppet, loses a game, assume that he loses with the lowest percentage of points possible, which is zero. This happens when his opponent, Tim Henman, wins the game to love, i.e.
MUPPET: 0 pts, HENMAN: 4 pts Muppet wins 0% of pts -- Ass. 1

Now let's consider a game which Muppet wins. Let's assume that Muppet wins the game to 30, so that the points scoring is as follows:
MUPPET: 4 pts, HENMAN: 2 pts Muppet wins 66.67% of pts -- Ass. 2

Note that this is not the lowest percentage of points which Muppet can win in a single game. For a number m deuces in the game, Muppet wins (4 + m) points and Henman wins (2 + m) points. As m tends to infinity (i.e. the game goes on forever), then the percentage of points won by Muppet tends to a lower bound of 50%.

Now let's consider a tie-break game which Muppet wins. The tie-break must be won with a minimum of 7 points, and by two clear points. Let's assume that Muppet wins the tie-break 7-5, i.e.
MUPPET: 7 pts, HENMAN: 5 pts Muppet wins 58.33% of pts -- Ass. 3

Once again, this is not the lowest percentage of points which Muppet can win in a single tie-break. Players can continue to win points alternately until Muppet wins with (n + 2) points, Henman having n points (where n > 5). As n tends to infinity (i.e. the tie-break goes on forever), then the percentage of points won by Muppet tends to a lower bound of 50%.

Now let's consider a set.
-------------------------

In a set which Muppet loses, he can maintain a zero percentage of points won by losing 6-0, each game lost to love as per Assumption 1 above. This gives a points tally:
MUPPET: 0 pts, HENMAN: 24 pts Muppet wins 0% of pts

Now consider a set which Muppet wins. If it is sets 1 to 4, then we could imagine he might win a low percentage of points if he wins by a score of 6-4, 7-5 or 7-6. Assuming that games are won as per the points totals in Assumptions 1 to 3, this gives:

Muppet wins 6-4:
MUPPET: 24 pts, HENMAN: 28 pts Muppet wins 46.15% of pts -- Ass. 4

Muppet wins 7-5:
MUPPET: 28 pts, HENMAN: 34 pts Muppet wins 45.16% of pts -- Ass. 5

Muppet wins 7-6:
MUPPET: 31 pts, HENMAN: 41 pts Muppet wins 43.05% of pts -- Ass. 6

Alternatively, in the fifth set, there is no tie-break and the set cannot be won 7-6. Instead players can continue to win games alternately until Muppet wins with (p + 2) games, Henman having p games (where p > 5). As p tends to infinity (i.e. the set goes on forever), then the percentage of points won by Muppet tends to a lower bound of 40%, which is lower than any of the set figures above.

[This figure of 40% becomes clear when you consider a pair of games, one won, one lost: total points scored in this pair of games is Muppet 4, Henman 6, total 4 + 6 = 10 points. Total points scored in the set is therefore: Muppet: 4(p + 2), Henman: 2(p + 2) + 4p]

Now let's consider the match.
-----------------------------

It is sensible to assume that Muppet will win with lowest percentage of points if he wins 3-2 in sets. Because of the effect of the tie-break, we initially need to consider two winning scores:

(a) 0-6 0-6 7-6 7-6 6-4
(b) 0-6 0-6 6-4 6-4 6-4

If the match is won by Muppet according to (a), then adding up the points tallies per set from Assumptions 4 and 6 gives:

(a) Muppet wins 0-6 0-6 7-6 7-6 6-4
MUPPET: 86 pts, HENMAN: 158 pts Muppet wins 35.25% of pts

(b) Muppet wins 0-6 0-6 6-4 6-4 6-4
MUPPET: 72 pts, HENMAN: 132 pts Muppet wins 35.29% of pts

Very close, but 0-6 0-6 7-6 7-6 6-4 gives the slightly lower percentage, so we can discard 0-6 0-6 6-4 6-4 6-4, and it looks like we have solved the problem: Muppet can win only 35.25% of the points if he wins 0-6 0-6 7-6 7-6 6-4.

Now let's check our assumptions. Firstly, let's consider a different score in the final set. Suppose the set goes on for longer, and Muppet wins 13-11, or (p+2) games to p. We calculated above that as p tends to infinity, the win points percentage of Muppet tends to 40%. As the score 0-6 0-6 7-6 7-6 6-4 gives a win points percentage lower than 40%, then adding games into the final set will bring the points percentage closer to 40% (effectively "drowning out" the effect of the 0-6 0-6 sets), i.e. making it higher.

Therefore we were correct to assume that the final set is won 6-4.

Now let's consider Assumptions 2 to 3, which relate to the length of the game. If a game goes to a number of deuces (or tied points in the tie-break), we calculated above that as m, n tend to infinity, the win points percentage of Muppet tends to 50%. As the score 0-6 0-6 7-6 7-6 6-4 gives a win points percentage lower than 50%, then adding deuces or extending the tie-break will bring the points percentage closer to 50% (again, effectively "drowning out" the effect of the 0-6 0-6 sets), i.e. making it higher.

Therefore we were correct to assume that no game goes to deuce and the tie-break is won 7-5.

And as for Assumption 1, that losing games are lost to love... well, if the game is lost by any other score then you are adding more winning points than you are losing points. Since our final percentage of 35.25% is below 50%, Assumption 1 must be correct. (Fairly trivial.)

So there you go. Possibly not a complete mathematical proof, but I'm pretty sure we've got the correct answer.

The answer is 35.25% and some Muppet beats Henman 0-6 0-6 7-6 7-6 6-4.

Answer: 35.25% (first correct solution: Clive)

Now for a puzzle which I'm not sure you can work out mathematically, but which is an interesting curiosity nonetheless (depending on one's interests, of course).

PUZZLE FOUR: MONOPOLY

I am playing a standard game of Monopoly based on the streets of London. (Go here if you want to check the layout of the board.) When I roll the two dice, by some sort of fluke, the total is always the same number (i.e. any number between 2 and 12, but the same number every roll).

Ignoring the effect of being moved around the board by Chance or Community Chest cards, there is/are certain square(s) on the board which is it impossible for me to land on, for any number I roll on the dice. Which square(s)?

Notes on the rules of Monopoly:

(a) if I roll 3 consecutive doubles (or land on the "Go to jail" square), I must go directly to jail. For the purposes of this puzzle, you may count this as having landed on two squares in the same turn i.e. the square that my third double takes me to (or the "Go to jail" square); and the Jail square itself.

(b) for the purposes of counting squares I've landed on, assume that Jail and the "Just visiting" part of the Jail square are the same square, and that if I land on either, you can count that square as one I've landed on.

Dinos Sfyris · Post by **Dinos Sfyris** » Mon Jan 04, 2010 6:01 pm

Question: After being sent to jail can you then roll a double or pay bail then continue round the board as normal?

If yes I think the only unreachable property is Mayfair

If no, 3 others can be added to the list.

Hugh Binnie · Post by **Hugh Binnie** » Mon Jan 04, 2010 6:27 pm

I got the same answer as Dinos. I've got working, if that's worth extra credit.

Dinos Sfyris · Post by **Dinos Sfyris** » Mon Jan 04, 2010 6:29 pm

I passed GO a bunch of times and collected £3600 altogether if that's worth extra credit.

Gavin Chipper · Post by **Gavin Chipper** » Mon Jan 04, 2010 6:29 pm

Gavin Chipper · Post by **Gavin Chipper** » Mon Jan 04, 2010 7:31 pm

Hugh Binnie wrote:I got the same answer as Dinos. I've got working, if that's worth extra credit.

I got the same answer as Dinos as well, but without any concerns about having to add three others to the list. I have confidence about what was meant.

Howard Somerset · Post by **Howard Somerset** » Mon Jan 04, 2010 7:37 pm

For me, the answer is just Mayfair.

For a bit of added info, throwing 9 each time gets you to the greatest number of squares (30), and throwing 10 each time gets you to the least number (4).

Nice puzzle. Wish I'd seen it before 7:15.

Matt Bayfield · Post by **Matt Bayfield** » Mon Jan 04, 2010 7:59 pm

Everyone who has posted an answer to the Monopoly puzzle is correct. Dinos responded first.

Matt Bayfield · Post by **Matt Bayfield** » Tue Jan 05, 2010 1:12 pm

ANSWER TO PUZZLE FOUR

To solve this problem it helps to look at the Monopoly board as 40 squares, where the Old Kent Road is square 1, and "Go" is square 40. Using this numbering basis, the two other squares of interest are square 10 "Jail" and square 30 "Go to jail". You start at "Go", on square 40.

You can now solve this problem by plotting out which squares you land on if you throw a 2 every turn, then a 3 every turn, and so on. You can stop plotting round the squares as soon as you arrive at a square you have already landed on.

Things to remember
------------------
(a) A throw of 2 or 12 will always be a double so you will automatically go to jail every 3 throws.

(b) For odd numbered totals on the dice, it is impossible to throw a double.

(c) For rolls of 4, 6, 8 or 10 it is possible to throw a double, and also possible not to throw a double. In these cases, first consider the squares landed on, starting as usual from "Go", as if no double is ever thrown. If, during this sequence, the "jail" square is never landed on, then also consider the different set of squares you would land on if you started from "jail" rather than starting from "Go". This covers the squares you would land on if you went to jail for throwing 3 consecutive doubles at any point.

Ultimately, you end up with a big table which looks a bit like the one below (where the numbers in the main section of the grid correspond to your turns e.g. the number “4” in row 12 in the column “dice roll of 3” indicates that you end up on the 12th square after 4 turns). You can see that only square 39 (Mayfair) cannot be landed on.

Code: Select all

  Square   ##        Dice roll ====>  ##       2       ##      3       ##       4                        ##       5       ##       6               7
           ## Is it a double? =====>  ##    Always     ##    Never     ##   Sometimes                    ##     Never     ##   Sometimes         Never
           ##                         ##               ##              ##  Start at Go  ## Start in jail ##               ##  Start at Go  ##
    1      ##                         ##               ##              ##               ##               ##               ##               ##
    2      ##                         ##             1 ##              ##               ##               ##               ##               ##           6
    3      ##                         ##               ##            1 ##               ##               ##               ##               ##
    4      ##                         ##             2 ##              ##             1 ##               ##               ##               ##
    5      ##                         ##               ##              ##               ##               ##             1 ##               ##          15
    6      ##                         ##   3 gotojail  ##            2 ##               ##               ##               ##             1 ##
    7      ##                         ##               ##              ##               ##               ##               ##               ##           1
    8      ##                         ##               ##              ##             2 ##               ##               ##               ##
    9      ##                         ##               ##            3 ##               ##               ##               ##               ##           7
 10 JAIL    ##                         ##    3 in jail  ##  10 in jail  ##               ##        START  ##             2 ##    5 in jail  ## 10 in jail
    11     ##                         ##               ##              ##               ##               ##               ##               ##
    12     ##                         ##             4 ##            4 ##             3 ##               ##               ##             2 ##          16
    13     ##                         ##               ##           11 ##               ##               ##               ##               ##
    14     ##                         ##             5 ##              ##               ##             1 ##               ##               ##           2
    15     ##                         ##               ##            5 ##               ##               ##             3 ##               ##
    16     ##                         ##   6 gotojail  ##           12 ##             4 ##               ##               ##             6 ##           8
    17     ##                         ##               ##              ##               ##               ##               ##               ##          11
    18     ##                         ##               ##            6 ##               ##             2 ##               ##             3 ##
    19     ##                         ##               ##           13 ##               ##               ##               ##               ##          17
    20     ##                         ##               ##              ##             5 ##               ##             4 ##               ##
    21     ##                         ##               ##            7 ##               ##               ##               ##               ##           3
    22     ##                         ##               ##           14 ##               ##             3 ##               ##             7 ##
    23     ##                         ##               ##              ##               ##               ##               ##               ##           9
    24     ##                         ##               ##            8 ##             6 ##               ##               ##             4 ##          12
    25     ##                         ##               ##           15 ##               ##               ##             5 ##               ##
    26     ##                         ##               ##              ##               ##             4 ##               ##               ##          18
    27     ##                         ##               ##            9 ##               ##               ##               ##               ##
    28     ##                         ##               ##           16 ##             7 ##               ##               ##             8 ##           4
    29     ##                         ##               ##              ##               ##               ##               ##               ##
30 GOTOJAI ##                         ##               ## 10 gotojail  ##               ##   5 gotojail  ##   6 gotojail  ##   5 gotojail  ## 10 gotojail
    31     ##                         ##               ##           17 ##               ##               ##               ##               ##          13
    32     ##                         ##               ##              ##             8 ##               ##               ##               ##
    33     ##                         ##               ##              ##               ##               ##               ##               ##          19
    34     ##                         ##               ##           18 ##               ##               ##               ##             9 ##
    35     ##                         ##               ##              ##               ##               ##               ##               ##           5
    36     ##                         ##               ##              ##             9 ##               ##               ##               ##
    37     ##                         ##               ##           19 ##               ##               ##               ##               ##
    38     ##                         ##               ##              ##               ##               ##               ##               ##          14
    39     ##                         ##               ##              ##               ##               ##               ##               ##
  40 GO    ##                         ##        START  ##       START  ##        START  ##               ##        START  ##        START  ##      START

  Square   ##        Dice roll ====>  ##      8                        ##    9     ##      10      ##    11     ##     12
           ## Is it a double? =====>  ##  Sometimes                    ##  Never   ##  Sometimes   ##   Never   ##   Always
           ##                         ## Start at Go  ## Start in jail ##          ##              ##           ##
    1      ##                         ##              ##               ##        9 ##              ##           ##
    2      ##                         ##              ##             5 ##       18 ##              ##           ##
    3      ##                         ##              ##               ##       27 ##              ##        13 ##
    4      ##                         ##              ##               ##          ##              ##         4 ##
    5      ##                         ##              ##               ##        5 ##              ##           ##
    6      ##                         ##              ##               ##       14 ##              ##           ## 6 gotojail
    7      ##                         ##              ##               ##       23 ##              ##        17 ##
    8      ##                         ##            1 ##               ##          ##              ##         8 ##
    9      ##                         ##              ##               ##        1 ##              ##           ##
 10 JAIL    ##                         ##              ##        START  ##       10 ##            1 ## 10 in jai ## 3 in jail
    11     ##                         ##              ##               ##       19 ##              ##         1 ##
    12     ##                         ##              ##               ##       28 ##              ##           ##          1
    13     ##                         ##              ##               ##          ##              ##           ##
    14     ##                         ##              ##               ##        6 ##              ##        14 ##
    15     ##                         ##              ##               ##       15 ##              ##         5 ##
    16     ##                         ##            2 ##               ##       24 ##              ##           ##
    17     ##                         ##              ##               ##          ##              ##           ##
    18     ##                         ##              ##             2 ##        2 ##              ##        18 ##
    19     ##                         ##              ##               ##       11 ##              ##         9 ##
    20     ##                         ##              ##               ##       20 ##            2 ##           ##
    21     ##                         ##              ##               ##       29 ##              ##        11 ##
    22     ##                         ##              ##               ##          ##              ##         2 ##          4
    23     ##                         ##              ##               ##        7 ##              ##           ##
    24     ##                         ##            3 ##               ##       16 ##              ##           ##          2
    25     ##                         ##              ##               ##       25 ##              ##        15 ##
    26     ##                         ##              ##             3 ##          ##              ##         6 ##
    27     ##                         ##              ##               ##        3 ##              ##           ##
    28     ##                         ##              ##               ##       12 ##              ##           ##
    29     ##                         ##              ##               ##       21 ##              ##        19 ##
30 GOTOJAI ##                         ##              ##               ## 30 gotoj ##  3 gotojail  ## 10 gotoja ##
    31     ##                         ##              ##               ##          ##              ##           ##
    32     ##                         ##            4 ##               ##        8 ##              ##        12 ##
    33     ##                         ##              ##               ##       17 ##              ##         3 ##
    34     ##                         ##              ##             4 ##       26 ##              ##           ##          5
    35     ##                         ##              ##               ##          ##              ##           ##
    36     ##                         ##              ##               ##        4 ##              ##        16 ## 3 gotojail
    37     ##                         ##              ##               ##       13 ##              ##         7 ##
    38     ##                         ##              ##               ##       22 ##              ##           ##
    39     ##                         ##              ##               ##          ##              ##           ##
  40 GO    ##                         ##       START  ##               ##   START  ##       START  ##    START  ##     START

[/size][/color]

Answer: one square - Mayfair (first correct solution: Dinos)

Puzzle five is on ten-pin bowling. Knowledge of the scoring system is required (as I don't have time to explain it right now - have to shoot off sharpish)...

PUZZLE FIVE: TEN-PIN BOWLING

Imagine you’ve just completed a normal 10-frame game of bowling. You notice that every time you bowled a ball at a full set of 10 pins, you managed to knock down a different number of pins (between zero and 10). What is the maximum score you could have achieved?

Kirk Bevins · Post by **Kirk Bevins** » Tue Jan 05, 2010 2:04 pm

Matt Bayfield wrote:
PUZZLE FIVE: TEN-PIN BOWLING

Imagine you’ve just completed a normal 10-frame game of bowling. You notice that every time you bowled a ball at a full set of 10 pins, you managed to knock down a different number of pins (between zero and 10). What is the maximum score you could have achieved?

So you could go 10, 9+1, 8+2, 7+3, 6+4, 5+5, 4+6, 3+7, 2+8, 1+9. Not quite sure how the scoring works...with so many spares and a strike to start with. Am I on the right lines or do have to hit a strike on the 10th ball, whereby giving me an extra ball or 2?

Innis Carson · Post by **Innis Carson** » Tue Jan 05, 2010 2:19 pm

Puzzle 5 attempt:

I can get up to 164 by doing 0/ X 1/ 3/ 4/ 5/ 6/ 7/ 8/ 9/2. The first ball, and the first ball thrown after the strike, don't count towards the score so you might as well stick 0 and 1 there. Don't think the rest of the order matters.

Hugh Binnie · Post by **Hugh Binnie** » Tue Jan 05, 2010 2:49 pm

Innis Carson wrote:Puzzle 5 attempt:

I can get up to 164 by doing 0/ X 1/ 3/ 4/ 5/ 6/ 7/ 8/ 9/2. The first ball, and the first ball thrown after the strike, don't count towards the score so you might as well stick 0 and 1 there. Don't think the rest of the order matters.

I imagine this is on the right lines but you knocked down the same number of pins with your second and third bowl.

Attempt: Using the same idea as Innis, I was able to get 162, bowling 1/10/2/3/4/5/6/7/8/9/0.

Matt Morrison · Post by **Matt Morrison** » Tue Jan 05, 2010 3:20 pm

With a quick attempt I've gotten 164

Ball 1 / Ball 2 / Score for end

1 / 9 / 10+9 (9 from the next ball as I've got a spare)
9 / 1 / 10+8
8 / 2 / 10+7
7 / 3 / 10+6
6 / 4 / 10+5
5 / 5 / 10+4
4 / 6 / 10+3
3 / 7 / 10+2
2 / 8 / 10+10 (10 from the next ball this time)
10 / - / 10+0+10 (0 and 10 from the next two balls as I've got a strike)
0 / 10 (have to start with a gutterball as have used all other numbers; doesn't contribute to score but does contribute to score gained from the previous strike in the 10th end)

19 + 18 + 17 + 16 + 15 + 14 + 13 + 12 + 20 + 20 = 164

Actually that doesn't work. Grr.

Jon Corby · Post by **Jon Corby** » Tue Jan 05, 2010 3:22 pm

Matt - you can't score two tens anywhere - the restriction is when you're facing 10 pins, NOT just on your first ball. I think Hugh's is right.

Matt Morrison · Post by **Matt Morrison** » Tue Jan 05, 2010 3:23 pm

Matt Morrison wrote:Actually that doesn't work. Grr.

Jon Corby wrote:Matt - you can't score two tens anywhere - the restriction is when you're facing 10 pins, NOT just on your first ball. I think Hugh's is right.

Beat you to it mate. Beat you to making me look stupid

Jon Corby · Post by **Jon Corby** » Tue Jan 05, 2010 3:25 pm

Matt Morrison wrote:
Matt Morrison wrote:Actually that doesn't work. Grr.

Jon Corby wrote:Matt - you can't score two tens anywhere - the restriction is when you're facing 10 pins, NOT just on your first ball. I think Hugh's is right.
Beat you to it mate. Beat you to making me look stupid

Bah, in that case I'll mock you for using the "/" all over your scoreboard, when that is the usual symbol for a spare. I found it very confusing

Howard Somerset · Post by **Howard Somerset** » Tue Jan 05, 2010 3:33 pm

My attempt:

In the 10 frames you get each possibility from 1 to 10, with the the second bowl, if there is one, clearing the rest. For the extra bowl after the 10th frame you get the 0.

1 and 2 are in the first frame and the frame following the 10. Otherwise the frames can be in any order.

The strike scores 20, the 10th frame scores 10, and the remaining frames score everything from 13 to 20, giving a total score of 162.

Matt Bayfield · Post by **Matt Bayfield** » Tue Jan 05, 2010 5:02 pm

Ooh - lots of attempts at this one... and I think Hugh, Jon and Howard are correct.

Having said that, for some reason the solution I've got stored on my computer from last year is different (and I have no idea why), so I'll check before posting the answer!

Matt Morrison · Post by **Matt Morrison** » Tue Jan 05, 2010 5:11 pm

Matt Bayfield wrote:Ooh - lots of attempts at this one... and I think Hugh, Jon and Howard are correct.

Having said that, for some reason the solution I've got stored on my computer from last year is different (and I have no idea why), so I'll check before posting the answer!

Haha, what the fuck! Jon gets counted as a correct answer simply for saying "Hugh's is right"? Blimey, the advantages of being in the clique!

FWIW, I did revise my working out as soon as I realised I couldn't knock over 10 after a gutterball, I just don't feel the need to post the same correct answer that's already been posted, albeit far more beautifully presented

Now praise me goddammit.

Matt Bayfield · Post by **Matt Bayfield** » Tue Jan 05, 2010 5:20 pm

Matt: well, denying Jon the opportunity to pick the hole in your original post, did make me smile, which has to be worth something. And of course praise is due for solving the puzzle, as indeed it is to all who get the correct answer, whether they post to say so or not!

I've made a mental note only to give credit for the first complete solution, in future.

Matt Morrison · Post by **Matt Morrison** » Tue Jan 05, 2010 5:25 pm

Hehe, just teasing obviously, will always take the opportunity to exchange pleasantries with Corby whenever possible.
I just didn't re-post as on my 2nd attempt I noticed that you can pretty much put the ends (it is 'ends' right, not 'frames'?) in almost any order to get 162.

Jon Corby · Post by **Jon Corby** » Tue Jan 05, 2010 9:44 pm

Matt Morrison wrote:Hehe, just teasing obviously, will always take the opportunity to exchange pleasantries with Corby whenever possible.
I just didn't re-post as on my 2nd attempt I noticed that you can pretty much put the ends (it is 'ends' right, not 'frames'?) in almost any order to get 162.

Nah, it's frames innit? Ends is the other sort of bowling, crown green and that.

Dinos Sfyris · Post by **Dinos Sfyris** » Wed Jan 06, 2010 8:25 pm

Matt Bayfield wrote:first correct solution: Dinos

GTFI! I like this first correct solution feature

Might I suggest all the winners get put into some bonus superdraw competition at the end or something?

5 awesome puzzles so far Matt, with thorough explanations and more mental twists than a Gordian knot. Keep em coming

In the last one, Like Innis I had 164, not realising after bowling a gutterball you'd again be faced with ten pins. Hope you don't mind if I set one of my own that I thought up on the tram home, inspired by your bowling puzzle.

PUZZLE D1: GOLF

There's a 9-hole golf course near my house where I like to play, but my golf swing is very inconsistent. One day while totting up the scores after playing with my friend, he points out to me that my difference in scores on subsequent holes are all different prime numbers. What is the lowest possible 9-hole total score I could have achieved?

Hugh Binnie · Post by **Hugh Binnie** » Wed Jan 06, 2010 8:41 pm

Dinos Sfyris wrote:PUZZLE D1: GOLF

There's a 9-hole golf course near my house where I like to play, but my golf swing is very inconsistent. One day while totting up the scores after playing with my friend, he points out to me that my difference in scores on subsequent holes are all different prime numbers. What is the lowest possible 9-hole total score I could have achieved?

How does a 9-hole golf course work? Does the par score have to be 36 or what have you? What's the highest par a hole can have?

Dinos Sfyris · Post by **Dinos Sfyris** » Wed Jan 06, 2010 8:47 pm

Hugh Binnie wrote:
Dinos Sfyris wrote:PUZZLE D1: GOLF

There's a 9-hole golf course near my house where I like to play, but my golf swing is very inconsistent. One day while totting up the scores after playing with my friend, he points out to me that my difference in scores on subsequent holes are all different prime numbers. What is the lowest possible 9-hole total score I could have achieved?
How does a 9-hole golf course work? Does the par score have to be 36 or what have you? What's the highest par a hole can have?

Par score doesn't matter. Although I probably should have mentioned the best possible score for any hole (however unlikely) is 1 ie a hole in one. Also to facilitate the puzzle there's no max stroke limit.

Howard Somerset · Post by **Howard Somerset** » Wed Jan 06, 2010 8:53 pm

I can't see any way of getting the 9 round score to be anything less than an amazing 247. Bul I'll keep trying.

Dinos Sfyris · Post by **Dinos Sfyris** » Wed Jan 06, 2010 8:57 pm

Howard Somerset wrote:I can't see any way of getting the 9 round score to be anything less than an amazing 247. Bul I'll keep trying.

Nope