A while ago one of our setters took Warfarin tablets every day. The dosage of these tablets is carefully monitored and managed, and the daily dose is quite precise. Once he had settled into taking the drug, the amount that he needed to take each day also settled, on 5.5 milligrams.
Warfarin tablets do not come in the 5.5 milligram size. In fact he received equal amounts of 1 milligram, 3 milligram and 5 milligram tablets. The tablets all had a groove in them which allowed them to be split in half. So, in effect he had 0.5 milligram, 1.5 milligram and 2.5 milligram tablets available as well.
It is easy to see that he could make his 5.5 milligram dose from one 5 milligram tablet and half a 1 milligram tablet. However this uses twice as many 5 milligram tablets as 1 milligram tablets, and uses no 3 milligram tablets at all.
As he had to pay for his medication he wanted to balance the usage of all three tablet sizes to that, over a period of time, he would use equal amounts of each tablet. And, of course, it would be wasteful to have unused tablets.
So, can you devise a sequence of tablets that delivers the 5.5 milligram dose every day, and uses the same number of each tablet? Because he did!
He used three different combinations to make 5.5 milligrams: the aforementioned one 5 milligram tablet and half a 1 milligram tablet; one and a half 3 milligram tablets and one 1 milligram tablet; and one 3 milligram tablet and half a 5 milligram tablet.
He took the first combination for ten days, using ten 5 milligram tablets and five 1 milligram tablets.
He took the second combination for six days, using nine 3 milligram tablets and six 1 milligram tablets.
This made a total of eleven 1 milligram tablets, nine 3 milligram tablets and ten 5 milligram tablets.
Two days of the third combination used two 3 milligram tablets and one 5 milligram tablet, making eleven of each size of tablet.
There might be other combinations and sequences, but that is the one he devised.