A Mathematical Buy!

In a classic wine shop in Flobecq, Belgium, list of three most popular wines are:

- The cost of 1 French wine bottle: 500$
- The cost of 1 German wine bottle: 100$
- The cost of 20 Dutch wine bottles: 100$

Homer Simpson entered the wine shop and he needs to buy

- All three types of wine shop.
- Needs to buy Dutch wine bottles in multiple of 20.
- Need to buy 100 wine bottles

Simpson has only 10000$. How many wine bottle(s) of each type, Simpson must buy? 

Simpson must buy.......Read More.... 

Source 
 

A Buy To Be Mathematical...

What was needed to buy?

Let's recollect the data where cost of each kind of wine bottle is listed.

- The cost of 1 French wine bottle: 500$
- The cost of 1 German wine bottle: 100$
- The cost of 20 Dutch wine bottles: 100$ (Cost of 1 bottle = 5$)

 

Now let F be the number of French bottle, G be the number of German bottles and D be the number of Dutch bottle that Simpson should buy.

F + G + D = 100

G = 100 - F - D   .....(1)

Total cost of all bottle must be $10000.

500F + 100G + 5D = 10000

Substituting (1) in above,

500F + 100(100 - F - D) + 5D = 10000

500F + 10000 - 100F - 100D + 5D = 10000

400F - 95D = 0

400F = 95D 
 
80F = 19D

D/F = 80/19

Possible values of D and F are 80 and 19 respectively.

From (1),

G = 100 - 19 - 80 =  1.

Let's verify if all these fits in his budget or not.  

19 French wine bottles would cost 19 x 500 = 9500, 1 German wine would cost = 1 x 100 = 100 and 80 Dutch wine bottles would cost 80 x 5 = 400. Remember we have got number of Dutch bottles in multiple of 20. Hence total cost = 9500 + 100 + 400 = 10000.

Hence with $10000, Simpson should buy 19 French, 1 German and 80 Dutch wine bottles if conditions of buying 100 bottles & Dutch bottles in multiple of 20 are applied.   

 

Test Of Poison

You are the ruler of a medieval empire and you are about to have a celebration tomorrow. The celebration is the most important party you have ever hosted. You've got 1000 bottles of wine you were planning to open for the celebration, but you find out that one of them is poisoned.

The poison exhibits no symptoms until death. Death occurs within ten to twenty hours after consuming even the minutest amount of poison.

You have over a thousand slaves at your disposal and just under 24 hours to determine which single bottle is poisoned.

You have a handful of prisoners about to be executed, and it would mar your celebration to have anyone else killed.

What is the smallest number of prisoners you must have to drink from the bottles to be absolutely sure to find the poisoned bottle within 24 hours? 

Here is the test designed for it! 

Source 

Test To Detect The Poison

Here is the challenge for us! 

Here binary number system can come to rescue. Just for a  moment, let's assume there are 15 bottles. Now let's number the bottles from 1 to 15. To test these 15 bottles we need 4 prisoners as below. Let's number the prisoners from in descending 4 to 1.

Wherever 1 is written for the particular bottle number, that bottle should be given to particular prisoner. Otherwise should not.

So for the specific bottle with unique number a specific combination of prisoners (they are bits here) would be formed. 

For example, if bottle labeled as 11 has a poison then prisoner no. 4,2,1 would die. In other words, if prisoner 4 & 2 die then the bottle no. 10 had poison.

For 16th bottle we would have needed 1 more prisoner.

In similar way, to test 1000 bottles, we need 10 prisoners (2^10=1024). Depending on what combination of prisoner die we can determine which bottle had poison. If prisoners numbered from 10 to 1 & if prisoner 10,8,6,3 & 2 die then bottle no.678 (binary - 1010100110) must had poison. Since the poison takes some time to take effect, even if prisoners taste this bottle, we still would have time to test rest of all bottles in given binary pattern. 

  Poisonous Bottle

In case there were 1025 bottle, we would have needed 11 prisoners.