## PRIZE PUZZLE FOR DECEMBER 2007## STAMP PUZZLEImagine that you have a computer program that tells you how many first and second class stamps to place on a package so that the excess postage is minimised. Actually, there is such a program that you can use and the link to it is HERE. Have a go. The default value of first class postage is 34 pence. The default second class postage is 23 pence. It was, and should be, 24 pence, but I have changed it to 23 pence for the sake of this puzzle. You can think about that and perhaps tell me why I found it necessary to change it. The stamp program is not essential to do this puzzle but it may help. Now here is the puzzle: You can only use stamps with a value of 34 and 23 pence which means to say that if you want, for example, to make up a postage of 97 pence, the best you can do is to use three 34-pence stamps which makes 102 pence. So your excess postage is 5 pence. Many values of postage will work out correctly and you will not have to pay more than the correct postage. In fact, for very high values it is guaranteed that every single value can be attained without any excess. So what is the highest value of postage that cannot be made exactly using those values of stamps? ## ANSWERThere is an interesting formula for this problem. It is(A-1)*(B-1)-1 where A and B are the stamp values without common factors. In this case A and B are 23 and 34 and the answer is 22*33-1 which is 725 pence. You were not expected to know the formula but there are other trial and error methods. I had to change the value of the second class stamp because if it remained at 24 pence (common factor 2) it would not be possible to place a limit on all the unobtainable odd values of postage. I received two correct answers, one from John Stafford and one from Christopher Broughton (my nephew, not a member) so there was no prize draw this month. |