Last week’s puzzle was getting U2 onto the stage in 17 minutes. The key insight is to realize that you have to get the two slowest people (Adam and Larry) to cross together, otherwise you haven’t a chance since the 10+5 minutes they use up separately doesn’t leave enough remaining time. But you can’t have them go first since there won’t be anyone to bring the flashlight back. Here’s the solution: Bono and Edge cross (2 minutes) and Bono returns (1 minute). Adam and Larry cross (10 minutes). Edge returns (2 minutes). Bono and Edge cross (2 minutes). Total 17 minutes.

Due the weakness of the bridge, a maximum of two people can cross at one time. To make matters worse, it is night-time and there is only one flashlight. The flashlight is always required when crossing the bridge and it must be walked back and forth, it cannot be thrown, etc. Each band member walks at a different speed and a pair must walk together at the rate of the slower man:

Bono takes 1 minute to cross
Edge takes 2 minutes to cross
Adam takes 5 minutes to cross
Larry takes 10 minutes to cross

For example, if Bono and Larry walk across first, it takes them 10 minutes to cross. If Adam then returns with the flashlight, a total of 15 minutes will have passed. How do they all get across in 17 minutes?

Last week’s puzzle was to find two US states that combined are anagrams of two other US states. It’s not quite a trick question but almost: the letters in North Dakota and South Carolina can (obviously) be rearranged to make South Dakota and North Carolina. And yes, it’s unique; unfortunately there isn’t a beautiful solution like Florida+Oregon = Nebraska+Ohio.

Last week’s puzzle was the cucumbers drying in the sun. There were 200 pounds of cucumbers originally and they were 99% water. That means that there was 198 pounds of water and 2 pounds of real cucumber stuff.

After sitting in the sun all day they were down to 98% water. The 2 pounds of real cucumber stuff was unaffected by this so that means that there were 98 pounds of water too. So the total weight was 100 pounds (half the amount started with). It’s counter-intuitive that going from 99% water to 98% water is actually a loss of over half the water.

Answer next week

Answer next week

Last week’s puzzle was the Monty Hall problem. It is sufficiently well-known and controversial that it has its own Wikipedia page. The reason it is controversial is that if it is expressed ambiguously then the answer is not clear and erudite professors will write in to complain that you’ve got it wrong. But as I expressed the problem (in the small print), you should definitely switch: you will double your chance of winning to 2/3.

The confusing thing is that intuitively it may seem like after Monty Hall has opened his door that your chances are 50:50 between the two doors. The right way to think of it is that the chance that the car is behind the door you picked starts off was 1/3 and that remains the case after Monty Hall has opened a door. After all, he was always going to open a door with a goat behind it (he’s not opening a random door that happens to have a goat behind it) since he knows where the goats are. So the chance that the car is behind the remaining door must be 2/3. It may help to think of the 100 door version. You pick a door, Monty Hall then opens 98 doors revealing goats, since he knows exactly which 98 doors to open and, except in the unlikely event you’ve actually picked the car, which is the one door he must avoid opening. The door he must avoid opening almost certainly has the car behind it.

So it plays out like this. The camel takes 1000 bananas to the first intermediate point 200 kilometers away. There it drops 600 bananas and returns with 200 (and it already ate 200). Again it takes 1000 bananas, drops 600 and returns with 200. Then it takes the last 1000 bananas, gets to the intermediate point, picks up another 200 (to replace the 200 it already ate) and takes them to the second intermediate point a further 333 1/3 kilometers away. There it drops 333 1/3 bananas and returns with 333 1/3 bananas. Then it takes a the remaining 1000 bananas from the first intermediate point to the second intermediate point, arriving with 666 2/3 bananas. It picks up the 333 1/3 bananas already at the second intermediate point (making 1000) and sets off for market 466 2/3 kilimeters away where it arrives with 533 1/3 bananas out of the original 3000.

Today’s puzzle: You are on Let’s Make a Deal. There are three doors. Behind one door is a car. Behind the other two doors are goats. You pick door number 1. Monty Hall, the host, who knows what is behind each door, then opens a different door to show you a goat. He offers you the choice of sticking with your choice of door number 1 or switching to the remaining closed door. Should you switch?

To make the problem unambiguous, assume that after you pick a door, Monty Hall must then open a door and he must pick a door that he knows has a goat behind it and that he must then give you the chance to switch.

Now for today’s problem. A banana plantation is located next to a desert. The plantation owner has 3000 bananas. The market is 1000 kilometers away. He has one camel which can carry a maximum of 1000 bananas at a time but eats one banana for every kilometer it travels. How do you get the maximum number of bananas to market?

Answer next week.

In any given year the weather station in New York’s central park records a certain rainfall. Ignore any trends and just assume that rainfall in one year is independent of rainfall the preceding year. A record year is one in which the rainfall exceeds that of any preceding year. Measurements started in 1835. In what year would you expect to get to the 20th record year (and as a hint, over the 160 year period from 1835 to 1994 there were 6 records).

Answer next week

This week’s puzzle is actually one that has some serious implications because it turns out that when it is given to physicians they typically get it completely wrong*

One in a thousand people have toenail cancer.  A screening test for toenail cancer is 95% effective, in that 95% of people with toenail cancer test positive, and 95% of people who don’t have toenail cancer test negative (there’s actually no reason in real life that for these two numbers to be the same).  In a routine screening for toenail cancer you are told you have tested positive. What is the probability that you actually have toenail cancer?

If you’ve not seen a problem like this before, take a guess (you’ll be wrong) before you try and actually work it out mathematically.

Answer next week

* See, for example, Gigerenzer, G. & Hoffrage, U. (1995). How to improve Bayesian reasoning without instruction: frequency formats. Psychological Review. 102, 684-701

Now Lisa would not be able to say she knew the number if she knew whether the first digit was a 3 or not unless it is that first group (4,16,36) and if she herself lives at either 4 or 16 leaving only two remaining choices. This means that I must have said untruthfully that my number was a multiple of 4 and truthfully that my number was a square. So we know that in reality my number is greater than 50, not a multiple of 4 and a square. The squares greater than 50 are 64, 81 and 100 and only 81 is not a multiple of 4.

I live at number 81.

We’re all too tired after DAC to have a really challenging puzzle. This week’s puzzle is to look at the following list of the numbers from 0 to 100 and fill in the missing piece in the midde:

8 18 80 88 85 84 89 81 87 86 83 82 11 15 50 55 54 59 … 22 2 0

Answer next week

What about going toward the whistle? That turns out to be a worst-case strategy since that is one place you can guarantee the train is never going to be again (unless the train is stationary or you are already on the track).

What about going directly away from the whistle? Again that is another worst-case strategy, guaranteed never to find the train (unless you are already on the track).

The best strategy, although it seems counter-intuitive at first, is to set off at 90 degrees from the direction of the whistle (either way is fine, it’s symmetric). If you find the track then wait. Maybe you chose the wrong direction and you never find the track, maybe the train’s already passed, maybe it was already going away from you when you heard it. But maybe you find the track and you get there before the train and it comes along and picks you up. This strategy maximizes that chance.

Today’s puzzle is another one that seems to have too little information until you think it through:

Both Lisa and I live on a street with house-numbers from 1 to 100. Lisa wanted to know at which number I live.

She asked me: "Is your number larger than 50?"

That seemed to make it a bit too easy so I lied.

Lisa next asked: "Is your number a multiple of 4?"

I was still in a bad mood so I lied again.

Then Lisa asked: "Is your number a square?"

I was feeling a bit guilty about all the lying so this time I told the truth.

Upon this, Lisa said: "I know your number if you tell me whether the first digit is a 3."

Well, I forget whether I lied or not this time. Lisa then told me where I lived. Of course she was completely wrong.

What house number do I actually live at?

Answer next week