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Puzzle for Today

Puzzle No. 442 – Friday 22nd March

My doctor prescribed some red tablets to be taken, one each day, from 1st January this year. They come in packets of ten. Later my doctor decided that I should also take some green tablets, one each day, from 1st February. The green tablets come in packs of seven. I am very conscientious about taking my tablets and I was wondering what would be the first date (if any) when I would be opening new packets of red and green tablets on the same day.

Today’s #PuzzleForToday has been set by Dr Lynda White of the Department of Mathematics at Imperial College London.

22nd March

On 31st January I would take the first of a new packet of red tablets, leaving 9 in the pack. Assuming the required date exists, let m be the number of packets of red tablets used after the 9 tablets in the opened packet have been taken, and let n be the number of packets of green tablets used from 1st February up to the day when packs of both colours finish. Then 9 + 10m = 7n.

Putting n = 1,2,3,4, 5 or 6 does not give an integral value for m but n = 7 gives m = 4. So on the 49th day (counting 1st February as day 1) I finish both a red pack and a green pack. This will be 21st March. The next day, 22nd March, I would open new packets of both colours.

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