Water and ..: Mathematical Paradoxes

A magic water-lily

There is a magic water-lily in a lake which can double its surface everyday. Therefore it can cover the whole surface of the lake in 20 days. How many days are needed so that the water-lily covers half the surface of the lake?

Answer

Don't be in a hurry to answer-think first!

You could start by wondering 1) about the growth rate of the water-lily. 2) which its initial surface was and more other details. Nothing, like that! Let's start all the way round! Think of the last day and the day before it!

On the 20th day the whole lake will be covered. On the 19th day, since it doubles everyday , the water-lily will have covered half the surface of the lake. You see! The has automatically been solved! In 19 days the water-lily can cover half the surface of the lake.

The newly weds and the crocodiles!

Three newly-wed couples were walking through the jungle when they came to a river which was very deep and wide.

To make matters worse, the river was full of hungry crocodiles. There was only one boat which could carry only three passengers at the time. But there were more to face. The three husbands were so envious that there was no chance to let their wives to be with another man either in the boat or anywhere else without their presence.

Try to find a way so that the three couples will be able to cross the river safely making the fewest possible boat-trips.

Answer

Let's use some symbols to make the problem easier to understand. We simply symbolize the three couples as Aa, Bb, and Cc. The capital letters correspond to the men where the small ones to the women.

At first, the three women use the boat to cross the river. Next, the woman "a" goes back and stays on the bank of the river until the three men, A, B, C cross the river . Five boat-trips were enough for all of them to cross the river safely.

The frog and the well

Persistently looking for water a frog fell accidentally in well which was 20 meters deep. Trying to get out of the well it followed some kind of a weird procedure. During the day it climbed three meters up whereas the night it fell 2 meters down. How many days on the whole did the frog to get out go the well?

Answer

May be someone could think that since the frog climbs 3 meters up in the day, 2 meters down in the night then in one whole day it goes 1 meter up. Therefore it will need 20 days not the whole.

But, let's go to the end of the 5th day, for example. How many meters has it moved up? Obviously 5 meters.. At the end of the 18th day? 18 meters, of course! That is the 19th day it will have covered three more meters, which means it will have managed to get out of the well. Someone may say that at the end of the 17th day it will have climbed 17 meters, consequently on the 18th day it will cover three meters more and will come out of the well. Wrong! It will be at the same height as the mouth of the well. Nobody can be sure it will jump out of the well!!!

Houses and taps

There are three houses and three taps. Try to connect the tap A with the house G, the tap B with the tap F and the tap C e connecting lines can be of any kind still they can't be intersected in any possible way, which means they shouldn't intersect another house or tap and also be out of the black frame. It's believed that there is no solution to this problem. What do you believe?

Answer

The solution is well presented in the sketch given below. What do you say now, can you connect every tap with any house? No matter how much you will try, you won't succeed. This problem has no solution!

A bath full of water

What weighs more:

a) a bath full of water

b) a bath full of water and a body floating on the surface or

c) both weigh exactly the same?

Answer

You may think that the bath with a body in it should weigh more. But in this case this bath contains less water than the one which is full of water. But can the quantity of water which is displaced by the body be made up for?

Yes, because the body displaces so much water as it is the body's weight, that's why the body floats. Therefore both weigh the same!

(This is a work of the Greek Comenius team)