Richard took glass, metal and plastic items to the recycling plant and received 1 "green point" for every 4 items in each category that he brought in. In any category, items in excess of a multiple of four were not counted, so, for example, if he brought in 21 glass items, 30 metal items and 43 plastic items, he'd earn [21/4] + [30/4] + [43/4] = 5 + 7 + 10 = 22 green points. (The [] square brackets indicate the floor function--the greatest integer not exceeding the value within.)

One week he brought in a 2-digit number of glass items, a larger 2-digit number of metal items and a still larger 2-digit number of plastic items. For each class of item he received a 1-digit number of green points.

Among the three 2-digit numbers and three 1-digit numbers involved, all the non-zero digits, 1 through 9, appeared exactly once.

If I told you the total number of items brought, you'd be able to deduce how many were glass, how many were metal and how many were plastic.

How many of each category were there?

(In reply to re: analytic solution by varadarajan)

 16,  29,  35 items for    4,  7,  8 points respectively, add up to  80 items.

But if I told you that there were 80 items, would you be able to know for sure that there were 16, 29 and 35 items? No, because there could have been  17,  25 and  38 items, with    4,  6 and  9 points and still have  80 items; or  18,  25 and  37 items for    4,  6 and  9 points, also totalling  80 items.

Only when there are 81 items can you uniquely determine from that fact alone that the numbers must have been the answer previously given. This is why the category is Logic, rather than Numbers.

Posted by Charlie on 2009-06-03 12:33:02