## Be Positive

Posted by rajesh on October 17, 2011 – 8:21 pm

Four positive integers which are different from each other total less than 18. To find out the four numbers you need to know their product and the smallest number.

But you don’t know their product and you don’t know the smallest number . . . or do you?

October 17th, 2011 at 10:26 pm

This one seems familiar . . .

Without working through all the possibilities it would seem that we need several combinations of numbers which have the same product, but only one of these has a unique smallest number.

The product 120 can be formed from 2345, 1456 and 1358. Only 2345 has a unique first number so I think that this is the one you want.

However, there may be other combinations with a different product that fit. I haven’t checked them all.

October 18th, 2011 at 5:53 am

Writing the possibilities out as ABCD,Sum,Product, we get

1 2 3 4, 24

1 2 3 5, 30

1 2 3 6, 36

1 2 3 7, 42

1 2 3 8, 48

1 2 3 9, 54

1 2 3 10, 60

1 2 3 11, 66

1 2 4 5, 40

1 2 4 6, 48

1 2 4 7, 56

1 2 4 8, 64

1 2 4 9, 72

1 2 4 10, 80

1 2 5 6, 60

1 2 5 7, 70

1 2 5 8, 80

1 2 5 9, 90

1 2 6 7, 84

1 2 6 8, 96

1 3 4 5, 60

1 3 4 6, 72

1 3 4 7, 84

1 3 4 8, 96

1 3 4 9, 108

1 3 5 6, 90

1 3 5 7, 105

1 3 5 8, 120

1 3 6 7, 126

1 4 5 6, 120

1 4 5 7, 140

2 3 4 5, 120

2 3 4 6, 144

2 3 4 7, 168

2 3 4 8, 192

2 3 5 6, 180

2 3 5 7, 210

The question implies that it is not sufficient to to merely know that the product is unique. As we’re in puzzleland, we can immediately remove all cases with a unique product, to get:

1 2 3 10, 60

1 2 4 9, 72

1 2 4 10, 80

1 2 5 6, 60

1 2 5 8, 80

1 2 5 9, 90

1 2 6 7, 84

1 2 6 8, 96

1 3 4 5, 60

1 3 4 6, 72

1 3 4 7, 84

1 3 4 8, 96

1 3 5 6, 90

1 3 5 8, 120

1 4 5 6, 120

2 3 4 5, 120

Only 2 3 4 5 can be singled out on the basis that it’ smallest member, 2, also has to be known. Just like wot Wiz said.

October 28th, 2011 at 2:59 am

The total combinations, if the sum of all the four digits are 18 is

I will tell you later the meaning of the last number.

1 2 3 12 — 8

1 2 4 11 — 7

1 2 5 10 — 6

1 2 6 9 — 5

1 2 7 8 — 14

1 3 4 10 — 16(not sure about this number)

1 3 5 9 — 16

1 3 6 8 — ?

1 4 5 8 — ?

1 4 6 7 — ?

So, according to question, there are 8+7+6+5+14+16+16+?+?+? combinations. And many of these combinations have same multiplication. So, i think one can’t find the correct answer if the smallest number with multiplication is not given.