Well done Andrei Lazanu, age 14, School 205 Bucharest, Romania and Robert Goudie, age 16, Madras College, Fife, Scotland. This is Robert?s solution.

It is possible to create all values above 29. This is because it is possible to create all values between 30 and 40, as shown below, and any other value greater than this can simply be created by taking the method for creating the value between 30 and 40 with appropriate value in the units column and adding an appropriate number of 10 lucres stamps. e.g. to create 77, we take the method for finding 37, which is two 6 lucres stamps and one of both the 10 and 15 lucres stamps and then adding 4 more 10 lucres stamps to bring the value up to 77. This works because our numbering system is base 10, and a 10 lucres stamp is available.

Value Number of 6 lucres stamps required Number of 10 lucres stamps required Number of 15 lucres stamps required Sum 30 0 3 0 0x6 + 3x10 + 0x15 = 30 31 1 1 1 1x6 + 1x10 + 1x15 = 31 32 2 2 0 2x6 + 2x10 + 0x15 = 32 33 3 0 1 3x6 + 0x10 + 1x15 = 33 34 4 1 0 4x6 + 1x10 + 0x15 = 34 35 0 2 1 0x6 + 2x10 + 1x15 = 35 36 1 0 2 1x6 + 0x10 + 2x15 = 36 37 2 1 1 2x6 + 1x10 + 1x15 = 37 38 3 2 0 3x6 + 2x10 + 0x15 = 38 39 4 0 1 4x6 + 2x10 + 1x15 = 39 40 0 4 0 0x6 + 4x10 + 0x15 = 40

Below 30, the pattern does not work, because there is no series of 10 consecutive values that are achievable.

The following numbers are the only values that are less than 30 that are possible.

Value Number of 6 lucres stamps required Number of 10 lucres stamps required Number of 15 lucres stamps required Sum 0 0 0 0 0x6 + 0x10 + 0x15 = 0 6 1 0 0 1x6 + 0x10 + 0x15 = 6 10 0 1 0 0x6 + 1x10 + 0x15 = 10 12 2 0 0 2x6 + 0x10 + 0x15 = 12 15 0 0 1 0x6 + 0x10 + 1x15 = 15 16 1 1 0 1x6 + 1x10 + 0x15 = 16 18 3 0 0 3x6 + 0x10 + 0x15 = 18 20 0 2 0 0x6 + 2x10 + 0x15 = 20 21 1 0 1 1x6 + 0x10 + 1x15 = 21 22 2 1 0 2x6 + 1x10 + 0x15 = 22 24 4 0 0 4x6 + 0x10 + 0x15 = 24 25 0 1 1 0x6 + 1x10 + 1x15 = 25 26 1 2 0 1x6 + 2x10 + 0x15 = 26 27 2 0 1 2x6 + 0x10 + 1x15 = 27 28 3 1 0 3x6 + 1x10 + 0x15 = 28 30 0 0 2 0x6 + 0x10 + 2x15 = 30

Hence the following numbers are the only numbers that cannot be created from stamps with values 6, 10 and 15:
1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 17, 19, 23, 29