Confusing Tablets

Ramaswamy is suffering from a disease. The doctor has given him two types of tablets in two different bottles. The two types of tablets are identical in all respects. He has to consume one tablet of one type and one tablet of another type daily. Else, he will die of over dosage. He can neither waste the tablets nor give them off. He took one tablet from one bottle on his hand and while tilting the other bottle, two of the tablets fell down and got mixed with the tablet of first bottle. Now he is confused between the three tablets.

Please help Ramaswamy so that he could manage consume one tablet of each type daily and save his life.


Hint 1

Make the problem simpler. Write down the given data and the problem statement neatly. This itself may lead to the solution.


Hint 2

Think of dividing or splitting each tablet.


Answer

Consider a tablet from the 1st bottle; add this to the 3 tablets. Split all the 4 tablets into 2 equal halves and place these halves in 2 different groups. Now the tablets in each group can be consumed by Ramaswamy daily.


Solution

We shall make the problem very simple. Let us name the 1st bottle as A and the 2nd as B. So, according to the problem, Ramaswamy now has 3 tablets in his hand i.e., 1 tablet from A and 2 tablets from B. We shall denote the tablets as (A, B, B).

Now Ramaswamy has to find a way so as to manage consuming the tablets (A + B) each day.

A small amount of thinking tells us that if we add another tablet A to the 3 tablets, we will have (A, A, B, B). With these tablets Ramaswamy can manage for 2 days consuming tablets (A + B) each day. But how do we separate A and B? We cannot recognize them.

This is the most important point from where the hint 2 will be very useful. It is not possible to recognize the tablets as A and B from the mixture. But we can divide or split each tablet to solve our problem. Consider the 4 tablets mixture (A, A, B, B). Now split each tablet into 2 equal halves and place them in 2 different groups.

Group 1	Group 2
A/2	A/2
A/2	A/2
B/2	B/2
B/2	B/2

Now each group has the tablets (A + B) which can be consumed for 2 days. This solves the problem.

If we think again and again about the solution we get many other solutions also e.g, for the 3 tablets (A, B, B), add 3 other tablets (A, A, B). We have (A, A, A, B, B, B). Now split each tablet into 3 equal parts and place them in 3 different groups

Group 1	Group 2	Group 3
A/3	A/3	A/3
A/3	A/3	A/3
A/3	A/3	A/3
B/3	B/3	B/3
B/3	B/3	B/3
B/3	B/3	B/3

Now each group has the tablets (A + B) which can be consumed for 3 days. Similarly many other solutions can be obtained.

Generalizing the solution, add (A,....,A,B,...,B) where A is repeated k times and B is repeated (k - 1) times where (k > 1) to the mixture (A, B, B) and split each tablet into (k + 1) equal parts and place them in (k + 1) different groups. Now each group has the tablets (A + B) which can be consumed for (k + 1) days.

Though all the above methods are solutions, majority of them are only theoretical, i.e, dividing each tablet into 3 or 4 or in general k equal parts is not practical. Hence we shall consider only the first solution as the perfect solution.