Jon Moon
Clarity and impact
Puzzles
Like puzzles? Try the ones below. They aren't to do with Clarity & Impact, but I like them. I hope you do too.Want more puzzles? Click here, scroll down, then enter your details. You'll receive a brief email every 7 weeks that'll occasionally include puzzles or other stuff such as charts that make me laugh - click here for examples.
Crossing a bridge in 17 minutes
Coins in bags
Two fuses
Two cars and a fly
Power plant and factory
Neil, Buzz and Michael's Moon rocks
Four men up to necks in sand
How many colours does a map need?
Crossing a bridge in 17 minutes
Four people have to cross a bridge. The bridge is weak, so they can only go two at a time. Also, it’s dark so they have to carry a torch back and forth – two cross with the torch, one returns with it, then two across again, etc. They walk at different paces: one takes 10 minutes to cross the bridge, one 5 minutes, one 2 minutes and one 1 minute. When people walk together, they walk at the pace of the slowest. E.g. the 2-minuter and 10-minuter take 10 minutes to cross together. How can they all cross in just 17 minutes (not 19)?[Bridge - answer]
Coins in bags
You've 50 bags of coins, and in each bag are 50 coins. All coins weigh 50 grammes - except for the 50 coins in one bag which each weigh 49 grammes. You've to find this one bag, and to help, you've a weighing machine… but you’re only allowed one reading from it. You can open bags, move coins about, etc... This is not a trick question.[Coins - answer]
Two fuses
The words below are slightly different to those in my December '09 update. Firstly, below it says: "you can move infinitely fast"; the answer explains why. Also, this version is even trickier... see the bit about "37 minutes and 11 seconds". Good luck, you will need it.
You have two one-hour fuses (the kind that often protrude from fireworks, not the electrical kind). Unfortunately, they don't necessarily burn at a uniform rate; all you know is that each one takes exactly one hour to burn completely. You can move infinitely fast. How can you measure 45 minutes? How can you measure 40 minutes? How can you measure 37 minutes 11 seconds? [Fuses - answer]
The "2 minute" and "1 minute" people go together, the "1 minute" person returns. The "10 minute" and "5 minute" people go across together, the "2 minute" person returns. The "2 minute" and "1 minute" people go across together. Total time: 17 minutes. The trick is for the two slowest to go across together.
Two cars and a fly
Two cars each drive at 10 mph towards each other. When they’re 20 miles apart, a fly sets off at 15 mph from one car to the other, and, on reaching it, flies back to the first car at which point it turns around and flies back to the other again, and so on. How far will the fly travel before it gets squished between two crashing cars?
Power plant and factory
On one side of a river is a factory. On the other side is a power plant, six miles downstream. The river is 0.5 miles wide. The factory needs power. It costs £16 per yard to run cable through the river and £12 along the riverbank. Where’s the best place to run the cable? This is not a trick question. It’s a maths riddle, and you need to know calculus to crack it. (PS I cribbed this riddle from a Maths book. Can’t remember which though.)
[Power plant - answer]Neil, Buzz and Michael's Moon rocks
Neil, Buzz and Michael return from a long trip with 100 valuable Moon rocks. But how to share them? They agree that Neil will suggest a split (e.g. 50, 25, 25), Buzz will make a counter-suggestion, and Michael will make a third suggestion. They will then vote on the suggestions and follow the majority vote. Assuming the rocks are unbreakable and equally valuable, that each man is intelligent, and that each wants to maximise his own winnings, what should the outcome be?
[Moon rocks - answer]Four men up to necks in sand
This one is really rather neat... click here for it (it's a bit longer than the others, so I've put it in a separate pdf document). I'll add the answer to this website in December 2010.
[Necks in sand - answer]How many colours does a map need?
Computers offer us millions of colours - but what's the minimum number of colours a map needs to ensure neighbouring territories aren't the same colour? Don't try to solve this using pencil, paper or PC, it's very tricky. Rather, it's just a curiosity to have a ponder on.
[Map colours - answer]
THE ANSWERS
Crossing a bridge in 17 minutes - answer
Coins in bags - answer
Take one coin from Bag 1, two from Bag 2, etc all the way to 50 coins from Bag 50. Put in a pile and weigh. If all coins were 50gm, the pile would weigh 63.75kg. The amount by which the pile is short of that tells you which Bag the 49-gm coins are in.Two fuses - answer
To measure 45 minutes, ignite both ends of one fuse and one end of the other. When the first fuse disappears, ignite the remaining end of the second. When the second disappears, 45 minutes have elapsed since the start.As for 40 minutes or 37 minutes and 11 seconds, below are the answers. I don't understand them. If you do, well done. I quote the answer in full.
To measure 40 minutes, ignite three flames and maintain three flames until both fuses are gone. 120 minutes worth of fuse divided by three is 40 minutes. Whenever two flames go out together (when a piece of fuse burning from both ends disappears) you must ignite two more to replace them, which you can do by cutting a piece in the middle and igniting both of the new ends (or just ignite the middle, which performs the cut for you). Whenever a single flame goes out (when a piece of fuse burning from one end disappears), you must ignite one end of another piece, which might require first cutting a piece if all pieces are burning from both ends. Notice that a potentially infinite number of operations may be required, so it's especially important that you move infinitely fast.
To measure any fraction n/d of an hour, where 0 < n/d <= 1, maintain d flames on the first fuse and d-n+1 flames on the second. When the first fuse is gone, extinguish all but one of the flames on the second, and maintain one flame. When the second fuse is gone, n/d of an hour has elapsed since the start.
Got it?!??!?!!
Two cars and a fly - answer
Fifteen miles. To get it, you needn’t muck about with algebra, sweating over sums. Rather, the cars travel at 10 mph towards each other and are 20 miles apart, so they will crash in exactly one hour. The fly travels at 15 mph, so it will travel 15 miles in that one hour. Simple, huh.Apparently, some people spend ages working out how far the fly travels each time in its backward-and-forward journey – then, after adding up ever-decreasing distances travelled by the fly, they eventually say the answer is “something like 14.96 miles”.
The short-cut approach is neater, quicker, simpler and more accurate.
Power plant and factory - answer
Run the cable from the factory to a spot on the other side of the river that’s 3/[2 x Squ Rt(7)] downstream…. To get this answer, you need to know Pythagoras’s theorem and know how to differentiate the square root of a (function of x). If you know these, you’ve probably already cracked the problem. If you don’t know them, telling you the answer won’t help you much. So I haven’t.Neil, Buzz and Michael's Moon rocks - the answer
(Thanks to Charles for sending me this. It's a fascinating puzzle because the answer seems just so counter-intuitive... and yet I can't fault the logic that derives it...)
Michael will get 99 rocks, and one of the others will get one.
In the following explanation, (x,y,z) means that Neil gets x rocks, Buzz gets y rocks and Michael gets z rocks. Of course, x+y+z=100.As the last player to propose, Michael is in a great position. He will try to cut out one of the others, and “team up” with whichever of the other two he can incentivise most cheaply.
If, for example, Neil offers (40,30,30) and Buzz offers (30, 35, 35), then Michael could incentivise Neil with 41 rocks or Buzz with 36 rocks. He’d opt for (0,36, 64) which would get two votes (his own and Buzz’s).
Neil and Buzz recognise this and will try to undercut one another. Buzz can always undercut Neil, except when Neil offers (1,0,99). In this case, Buzz can only offer (0,1,99) and hope that Michael votes for this rather than for Neil’s offer.
(Note that any attempt by Neil and Buzz to co-operate can be outbid by Michael. If Neil proposes (50,50,0) and Buzz makes an identical counteroffer, then Michael can offer (51,0,49). No good for Buzz, so Buzz would undercut Neil. Neil’s attempt at co-operation is doomed.)
Four men up to necks in sand
Answer: C. D’s silence gives C the answer – you see, C can see that B has a white hat, and realises that, if C was also was wearing a white hat, D would work out that his own hat must be black and so could then shout out an answer. However, D doesn’t shout out an answer… so C knows his own hat must be different to B’s. After a minute or so of silence, C therefore yells out “my hat’s black”. And everyone lives.How many colours does a map need?
Just four. Consider America: with four colours, you can colour each state without neighbouring states having the same colour.Long ago, people realised four was always enough for any map - but proving it was extremely difficult. From 1852, this ‘four-colour’ problem was one of the great unsolved proofs of maths. Then in 1976, two people proved it by getting a computer to check thousands of exceptional cases. It was an inelegant and controversial proof but is still the only proof to date though.
And what’s this to do with clarity? Don’t go overboard with fancy colours
