A Butterfly and 2 Bicycles (Solution 1: 10 Years, Solution 2: 19 years)

This is a fun math problem with at least 2 ways of solving it the simpler way people call the "Trick".

A girl and a boy are moving between each other at 10 km/hour starting 20 km apart. There is a butterfly which moves between them at 5 km/hour. How far does the butterfly have to fly?

There is a popular legend about the great mathematician John von Neumann that he solved this problem in a few seconds but not by using the trick answer but by doing it the hard way.

See if you can do it faster than von Neumann. First with the trick and then the hard way. (Hint: The hard way sums an infinite series)

A Simple Game: 8 Years

OK!! I guess you know the game of scissors, paper and rock.

Well this is a little bit simpler because instead of having 3 things to choose from you choose between rock and flower. We will say having your hand open is the sign for a flower and having your hand closed as before is the sign for the rock.

Now what is more difficult with this game is that the rewards depend as follows:

If both payers show flowers you both get 10 marshmallows. If both players show rocks you both get 5 marshmallows. If one of you shows a flower and the other shows a rock only the rock gets 20 marshmallow the flower gets none.

Assuming that you have no idea what the other player will do what is your best thing to show to get the maximum amount of marshmallows?

We call the numbers:

• 5 the number for being tough.
• 10 when everybody is nice.
• 20 for being tough when the other is nice.
• 20 for being nice with the other is tough.

What happens if we know the other player always seems to choose the flower?


These types of games are taken from a field of mathematics called Game Theory and are used in everything from business to politics.

The Swimming Pool Business

You are visiting your grandparents in the countryside.  And it is a really warm summer but there are no lakes and the they live far from the seaside.  

One extreamly sweltering day you are with your friends thinking about how to stay cool in the summer.

One of your friends mentions that there is a house hidden away on the mountain slope which was built some 50 years ago with a swimming pool.  The pool is still in pretty good shape but it needs to be cleaned and filled with water to use.

It turns out one of your friends grandmother lives in the house but doesn't have the money nor the interest in swimming to keep the pool in working condition.

So you sit down with your friends and do the calculations of what it would cost to operate the pool and how much you would have to charge per visit per day per person.

First you have to fill the pool.  The pool is 5 meter long, 4 meters wide and 1 meter deep.  Your dad says that water costs 5€ per cubic meter.  During the summer months you will need about 10€ of chemicals to keep it nice and blue.  

Your grandparents aren't too excited about having a whole bunch of kids in the pool without a lifeguard so they say that you need to have a lifeguard to operate the pool.   You immediately think of Bruce the visiting swimming champion from Australia who is visiting the town this summer.  Bruce tells you that he would be happy to work as a lifegaurd if he can use the pool for free and you pay his lunch at the organic supermarket every day he works which cost 5€.

As a favor to your friends Grandmother for letting you and your friends use the pool for the summer you offer to paint her house for free.  The paints cost 250€.

If you run the pool everyday during the summer for 3 months and there are on average 20 visitors every day how much do you need to charge?

Multiplication and the Multiplier Effect! Secrets about Banks! : 10 Years

Banks do something really special! 

Because banks are insured by the government (for the most part) it makes them pretty safe places to store your money.

Now when you do that, there is a curious thing that happens.  To you, your money is still there, because if you look in your bank account online you will see it or if you call your banker and ask him he will say it is there.

But you know what?  It actually isn't there.  The Bank has taken your money and loaned it out to somebody else.

That's right those dollar bills you handed the teller are not there anymore.  

But don't get worried or call the FBI- it the way the modern economy is supposed to work.  Your money is being used to do something valuable like build a dam, a ship or something else that will generate profits so that the people and businesses who built these things can pay the bank back your money and the interest you are earning and a bit of interest to the bank too.

Now lets say somebody borrowed the money you put in the bank to make a bridge.  

They take the money and go to a construction company and pay for the bridge to be built.

But guess what - the construction company has a bank account at your bank too.  So they put the money in there until the start building.

And then along comes somebody who want to build a building and the borrow that money to make a building.  

Oh and just for fun lets say its the same construction company.  So they take that money and put it in the bank too.

So here comes the question:  There are 1000 individual savers like you who each put $100 in the bank. How much money can we loan the people making the bridge and the people making the building?  

OH there is one more thing - the Federal Government used to make sure than banks had to keep at least 10% of its money with out loaning it out (it no longer requires that, which is probably too bad).

If you explain to people how banks work they will be really impressed and probably recommend you to go to work for the Federal Reserve.

Oh you can call it the money multiplier effect for more...well effect!!

How Banks Work: 8 years

Banks allow you to save your money in a relatively safe place since the deposits are insured by the government (the same people who make the money in the first place) and in return for saving your money in the bank you are paid interest.  Interest is usually calculated in percent paid per year and often banks pay 3-7% interest on the money you loan them.

To make a profit banks take your money you have saved with them and loan it out to other people charging more than they pay you.

Lets say that you save 100 €  at a bank and the bank pays you 2%. 

How much does the bank pay you at the end of one year (don't include your original 100€)?  This is the bank's cost.

Now lets say the bank loans out 80 € of your money to somebody else at 8% how much do they earn from the person taking the loan.  This is the bank's income.

Since 

Profit = Income - Cost

How much is the bank's profit from your 100€ (excluding taxes and other costs)?

The Newspaper Business: 7 years

You are the publisher of a newspaper and every week you sell one hundred copies of your newspaper.  You have to pay for the articles and you get paid for the ads according to the image above.  

In addition you charge 25 cents for the newspaper but on the other hand it costs you 5 cents to print.

How much money do you make a week if you dont have to pay taxes or other costs?

Making Windmill Coils: 7 years

For this problem you need to remember that the formula for the circumference of a circle is 2*Pi*r.

You are in the business of making windmills. 

 One important part of the windmill which allows it to make electricity are the coils. There are normally several coils in a generator.  For the model your company is making there are 36. 

 To make a coil we wind copper wire around something to make it look like this:

When you look at the coil from one side you see that is circular.

The letter r represents the radius and lets say that for our coils it is 2 cm.  

If we need 36 coils with with 1oo windings or loops each and each meter of copper wire costs 10 cents.  How much do the coils for a windmill cost to make?