Using numbers from 1-5, how many 4 digit combinations add up to 10?

Answer 1

You can reason this out without any fancy math ... try to kinda count using only the digits 1 - 5 and the numbers have to be at least 4 digits long.

The smallest 4 digit number I know is 1000 and the largest is 9999 ... However, these are not allowed ... so let's start counting from 1000 to 9999 skipping those numbers that don't only have the digits 1 -5 in them (mind the total of the 4 digits while you're at it and see if you notice a pattern).

number total

1111 4 (notice I skipped all the numbers from 1000 to 1110 because they have digits other than 1-5)

1112 5

1113 6

1114 7

1115 8

1121 5 (once again we skip over all the numbers that contain digits other than 1-5)

1122 6

1123 7

1124 8

1125 9

1131 6

1132 7

1133 8

1134 9

1135 10

1141 7

1142 8

1143 9

1144 10

1145 11

we see a pattern developing ... our first number where the digits total 10 is 1135, then 1144, so we will get one and only one every decade (say from 1131 to 1135, or 1141 to 1145 or 1151 to 1155, etc.) until the total of the first number of the decade is greater than 10:

like this:

1135, 1144, 1153,

1225, 1234, 1243, 1252,

1315, 1324, 1333, 1342, 1351,

1414, 1423, 1432, 1441,

1513, 1522, 1531,

(19 numbers above)

2125, 2134, 2143, 2152,

2215, 2224, 2233, 2242, 2251,

2314, 2323, 2332, 2341,

2413, 2422, 2431,

2512, 2521,

(18 numbers above)

3115, 3124, 3133, 3142, 3151,

3214, 3223, 3232, 3241,

3313, 3322, 3331,

3412, 3421,

(14 numbers above)

4114, 4123, 4132, 4141,

4213, 4222, 4231,

4312, 4321,

4411

(10 numbers above)

5113, 5122, 5131

5212, 5221

5311

(6 numbers above)

19 + 18 + 14 + 10 + 6 = 67 numbers

Done!

(somebody want to check me ;-) !! have fun!!)

(this would also make a good computer programming problem ... can you imagine the algorithm?)