There's something special about this set...

Probability Level 3

How many different integers can be expressed as the sum of three distinct members from the set ${{1, 4, 7, 10, 13, 16, 19}}$ ? Try to find a creative way to solve this!

The answer is 13.

3 solutions

Daniel Liu
Apr 10, 2014

Note that the sum must be divisible by three, because the numbers are all $1\pmod{3}$ . The smallest possible sum is $1+4+7=12$ , and the largest is $13+16+19=48$ . Between $12$ and $48$ inclusive, there are $13$ integers divisible by $3$ . Therefore, our answer is $\boxed{13}$ .

Beautiful! That's exactly the solution I was looking for! Now, I encourage others to find an interesting approach to solve the problem. :D

Finn Hulse - 7 years, 2 months ago

Finn, I arrived at the solution by defining a set equivalent to the original one as follows. Let a=1, so the set is {a,a+3,a+3+3,....,a+3+3+3+3+3+3}. Therefore , the lowest unique integer that could be produced by this set is 3a+3+3+3=12 and the maximum unique integer is 3a+3+3+3+3+3+3+3+3+3+3+3+3+3+3+3=48. From this you can deduce that there must be 13 unique integers produced by this set since every succeeding unique integer after the lowest one increases by a common difference of 3. Thought this would be a different way of looking at hopefully... but one of my friends actually found a more interesting way using graph theory!

Adrian Castro - 5 years, 12 months ago

Great solution!

Mahdi Al-kawaz - 7 years, 2 months ago

Awesome solution!!!!!

Adarsh Kumar - 7 years, 2 months ago

Same solution as mine but I overlooked that there are 13 integers which are divisible by 3 between 12 and 48

Jayver de Torres - 7 years, 1 month ago

Pradeep Ch
Apr 26, 2014

the given sequence is an AP. so, any sum of any three numbers will be of form: 3a + (l+m+n)d where a is first term and d is the common difference. and l,m,n will be from 0 to 6. now, number of different combinations of l+m+n can be found out as : minimum value of l+m+n = 0+1+2 = 3 and maximum = 4+5+6 = 15. so, there will be total of 13 different combinations of l+m+n and hence 13 different integers. :)

Lokesh Sharma
Apr 11, 2014

Python's Solution:

import itertools

listOfNumbers = [1, 4, 7, 10, 13, 16, 19]
allCombinations = list(itertools.combinations(listOfNumbers, 3))

possibleIntegers = []
for i in allCombinations:
    if sum(i) not in possibleIntegers:
        possibleIntegers.append(sum(i))

print len(possibleIntegers)

Advantage: It doesn't require any thinking.

Cool!

Finn Hulse - 7 years, 2 months ago

There's something special about this set...

The answer is 13.

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