These mini-puzzles are a small collection of teasers to make you think. Don't reveal the answers until you've thought about them for a bit!

TV game show

Imagine you're on a game show on tv, and you're offered the choice of three prizes behind closed doors. One of the prizes is very valuable (usually a car), and behind the other two doors are worthless prizes (often goats). You don't know what's behind which door, but the game show host does. You choose a door (hoping to get the car). The host, instead of opening your selected door, raises the tension by opening one of the other two doors instead, behind which he knows there is a goat. Then he offers you the choice - do you stick with your previously-selected door, or do you switch to the other, still closed, door?

Answer: You raise your chances of getting the car by switching. The chance of the car being behind your original choice is 1 in 3. Now that one of the other two doors has been removed, the chance that the car is behind the other closed door is now 2 in 3, which is over twice as likely. It's extremely counter-intuitive because it's common to think that the chances are now 50:50, but they're not. Another way to look at this: if you chose a goat, you should switch, and if you chose the car, you should stick. But in the first round, you probably chose a goat (which was twice as likely as choosing the car). So you will probably gain by switching doors, and probably lose by sticking with your original choice.

For more information on this so-called "Monty Hall problem", see Wikipedia. There's also a clear description of the solution in the popular book The curious incident of the dog in the night-time.

Average speed

You plan a bicycle journey, from your home to the next town and back again on the same route. You would like to get an overall average speed of 20 km/h. After setting off, you get delayed, so that when you reach the town, your average speed is only 10 km/h. How fast do you have to cycle on the return journey so that your average speed for the whole trip is 20 km/h?

Answer: You can't do it. In order to get an overall average of 20 km/h, you would need to double the average of the outward journey. You can only do this if you cover the return distance in zero time, which is not possible.

One common answer is 30 km/h, so that the average of 10 and 30 is 20, but the speeds do not get averaged in this way. If we let the distance to the town be d, then the time taken for the outward journey is d/10 hours. If the return speed is 30 km/h, then the time taken is d/30 hours. So the overall speed is 2d divided by the total time (d/10 + d/30) which gives 15 km/h, still slower than the 20 km/h you wanted.

To get 20 km/h, you would need to cover the return distance d in a time t, where the average speed is 2d divided by (d/10 + t). This is only possible if t equals zero, which means you've used up all the available time on the outward journey and however fast you cycle back, your average will always be less than 20 km/h.

Hot coffee

You make a cup of coffee, but before you add the milk you realise that it's too hot to drink. If you want the coffee to cool down to drinkable temperature as soon as possible, should you add the milk now or later?

Short answer: Later. If you add the milk now, it will reduce the temperature, but this means that the coffee will lose heat more slowly to the air. If you wait, the coffee will remain hotter for longer, losing heat more rapidly to its surroundings, so that when you do add the milk, the combined temperature will be lower.

Obviously there are exceptions to this rule, such as when adding the milk immediately would bring the temperature down to a drinkable level straight away. And you can make it more complicated by considering the rise in temperature of the milk (if it's out of the fridge) while you're waiting. But the simple answer is often realistic and usually counter-intuitive.

Numbers

This one is more mathematical. You start with three numbers, and you try to make the number 6 by drawing mathematical symbols between the numbers to make a sum. What is a mathematical symbol? Well, for starters there are the normal add, subtract, multiply and divide. For some you might need to add brackets ( ), and for the harder ones you might need to resort to additional operations. But you can't draw any extra numbers in there, only symbols.

Here are the sums, and the first one is done for you.

Medium answers:
Numbers 4 and 9 use the square root symbol and reuse the simpler answers for 2 and 3; numbers 5 and 7 require brackets:
√4 + √4 + √4 = 6
5 + (5 / 5) = 6
7 - (7 / 7) = 6
√9 * √9 - √9 = 6

Difficult answers:
Number 1 requires a factorial symbol !, and 8 requires two square roots:
(1 + 1 + 1)! = 6
8 - √(√(8 + 8)) = 6

An alternative solution for number 8 uses the logarithm to base 2, to give log_{2}(8) * log_{2}(8) - log_{2}(8), but that doesn't work because you're not allowed to write the additional 2. In German, it works, because the symbol is lb. Similarly, you could use a cube root of 8 to give 2, but the standard symbol for cube root includes the number "3", so that's also out. Also, number 10 can be solved using logarithms to base 10, but only if you don't have to write the "10" in the symbol.