# Jon Moon

## Clarity and Impact

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# 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?

Multiplication puzzle

Four cards on a table

Next number in sequence

The 2015 Singapore maths puzzle

Dropping anchor

Condemned man's marbles

Cycling and wind resistance

Blindfolded person and cards

Two numbers in sequence

## 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]

## 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).

[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]

## Multiplication puzzle

This represents the multiplication of a 4-figure number by 3:

ABCD x 3 = EFGHJ.

A to J are all single numbers. The calculation uses each of the digits 0 to 9 once and once only (3 is already used, obviously). The 4-figure number contains three consecutive numbers, which are not in order, and its third digit, C, isn’t one of the consecutive numbers but is the sum of two of those consecutive numbers. As for the 5-digit number, E, G and J are three consecutive numbers, again not in order. And F and H are also consecutive numbers. What are all the numbers?”

[Multiplication puzzle - answer]

## Four cards on a table

You've four cards on a table in front of you, and you know that each has a number on one side and a letter on the other. You see the letters D and K, and the numbers 3 and 7 staring up at you from the four cards. What is the fewest number of cards you need to turn over to check the claim that: "Every card that has a D on one side has a 3 on the other"? And which cards?

[Four cards on a table - answer]

## Next number in sequence

My son showed me this. What's the missing number in the sequence?

06 68 88 XX 98

It stumped me. My wife got it in less than two seconds.

[Next number in sequence - answer]

## The 2015 Singapore maths puzzle

In April 2015, this puzzle went viral. It was set by a teacher in Singapore to a bunch of 14/ 15 year olds. Not average ones though - the top ones. It captured the imagination of the world – and it’s great. Enjoy.

Albert and Bernard just become friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates: May 15, 16, 19, June 17, 18, July 14, 16, August 14, 15, 17. Cheryl then tells Albert and Bernard separately the month and the day of her birthday respectively. (Editor: that is, Albert gets told one of May, June, July and August, and in a separate conversation, Bernard gets told one of 14, 15, 16, 17, 18, 19.)

Albert: “I don’t know when Cheryl’s birthday is, but I know that Bernard doesn’t know too.”

Bernard: “At first, I don’t know when Cheryl’s birthday is, but I know now.”

Albert: “Then I also know when Cheryl’s birthday is”

When is Cheryl's birthday?

[Singapore maths puzzle - answer]

## Dropping anchor - does water rise or fall?

You’re in a boat on a lake and drop anchor. Does the water level of the lake go up, down or stay the same?

## The condemned man's marbles

You’re condemned to die, but have a chance of a pardon.

You’re handed a bag of 50 black marbles, a bag of 50 white marbles and two bowls. You must pour all the marbles into the bowls, distributing them between bowls as you wish. You’ll then be blindfolded, the marbles swirled within each bowl, and bowls may even be switched around. You then must touch the outside of one of the bowls and take a marble from the bowl you touched.

If the marble is black, you die. If white, you’re pardoned.

How should you distribute marbles to give you the best chance of living?

[Marbles puzzle - answer]

## Cycling and wind resistance

I cycle 12km to work. If the wind is against me, I take 40 minutes. If it’s with me, I take 30 minutes. How long do I take if there’s no wind?”

## Cards and the blindfolded person

You're wearing a blindfold, and are handed a deck of 52 cards, and are told ten of the cards are face up, the rest face down. How can you divide the cards into two piles and ensure that each pile has the same number of cards facing up?

[Cards - answer]## Two numbers in sequence

## 61 52 63 94 ?? 18 - what is the '??' number?

[Two sequence - answer]

Four squaresClick here for a PowerPoint file (on the SlideShare website) that has both the questions and answers for a series of puzzles. I didn't solve the fourth one, let alone do it in the 'World Record' time of seven seconds. Good luck. (PS: after the fourth puzzle, there are some explanatory words that might or might not float your boat.)

THE ANSWERS## Crossing a bridge in 17 minutes - 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.

## 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

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? - answer

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

## Multiplication puzzle

The three consecutive numbers must be 0,1,2 or 4,5,6, or 5,6,7, etc. However, ‘3’ is already accounted for. But we know the third digit, C, is the sum of two of the consecutive numbers.

So, if the consecutive numbers are 0,1,2, we have a problem: the sum of any two consecutive numbers can only 1,2 or 3, but we’ve already accounted for 0,1,2 and 3.

So the consecutive numbers can’t be 0,1 or 2. But they can’t be 5,6,7 or anything higher – the sum of two digits is greater than 9. So the consecutive numbers are 4,5,6. The third digit is therefore 4+5, i.e. 9. ABCD contains 4,5,6,9, but in what order? D can’t be 5, because J would then also be 5.

Also, we know C is 9, so CD is either 96 or 94. But 3 x 96 is 288 – and we can’t have ‘8’ twice. So CD is 94. So HJ is 82. F is 7 (consecutive to 8). E can’t be 0, so it’s 1. G is 0.

EFGHJ is therefore 17082. ABCD is 5694.

Got that?

## Four cards on a table - answer

Two cards - the D and 7. You need to turn over the D card and don't need to turn over the K card (they're the easy ones). You needn't turn over the 3 card because the claim isn't: "Every card with a 3 on one side has a D on the other". As for the 7 card, you need to check it doesn't have a D on its other side.

## Next number in sequence - answer

The intended answer is: 87. Swivel your head and read the numbers upside down and the’re 86 XX 88 89 90. Unwittingly though, there’s another answer, albeit it’s less elegant: 89. The first digit of a number if the second digit of the prior number. This unwitting alternative doesn’t arise if the puzzle is reversed, i.e. 98 XX 88 68 06. Maybe that should have been the puzzle.

## The 2015 Singapore maths puzzle - answer

Firstly, if Bernard doesn’t know, the birthday can’t be May or June, for they’ve dates that no other month has (19, 18). That is, if Cheryl had said ‘May’ to Albert, Bernard might have been told ‘19’ – in which case Bernard would definitely know the birthday is May 19, for there is only one ‘19’ date on offer. But Albert knows Bernard doesn’t know. So it’s not ‘19’ nor ‘18’ – hence no May or June.

So far, so good. Bernard works this out too – he hence knows it’s not May or June, but July or August. And then says he now knows the birthday. So it can’t be a ‘14’ birthday, for he wouldn’t know if it was July or August. So it must be either July 16 or August 15 or 17.

Albert work this out too. And he then say he now knows the birthday. Which means Albert couldn’t have been told August, for he wouldn’t know if it was August 15 or 17. The answer is hence July 16.

## Dropping anchor - does water rise or fall? - answer

The water level goes down. While the anchor is in the boat, it displaces an amount of water equal to its weight. When it is dropped overboard, the anchor displaces an amount of water equal to its volume.

## The condemned man's marbles - answer

You should put one white marble in one bowl, and the rest in the other. That way, you're pardoned if you choose the 'one white marble' bowl, and have a 49/99 chance if you choose the other bowl. Overall chance of survival: nearly 75%.

## Cycling and wind resistance - answer

If I ride 12 km in 30 minutes with the wind, I ride 16 km in 40 minutes with the wind. Against the wind, I ride 12 km in 40 minutes. So I ride 14 km in 40 minutes if no wind. I need to ride 12 km, so it takes me (40/14) x 12 minutes, i.e. 34 minutes and 17 seconds.

## Cards and blindfolded person - answer

The person divides the cards into piles of 42 and 10, then turns the whole pile of 10 over.

## Two sequence - answer

?? is 46. Each pair of numbers is a square, reversed - that is, 61 = 16, 52 = 25, 63 = 36. The missing pair is 8 squared, i.e. 64, reversed to 46.