What's in my pocket? [duplicate]

Question

Well, I can tell you Johnny has memory cards in his pocket.

Back Story

My brother, Johnny, is a tech nerd. He loves gadgets of all kinds. As a matter of fact, you can be sure at any one given time, he will have SOME sort of tech related thing in his pockets.

That being said...

Johnny-Gadget says to our other brother, Mickey, "Can you figure out how many memory cards I have in my pockets?"

He then gives Mickey three clues:

1. If the number of memory cards I have is a multiple of 5, it is a number between 1 and 19.

2. If the number of memory cards I have is not a multiple of 8, it is a number between 20 and 29.

3. If the number of memory cards I have is not a multiple of 10, it is a number between 30 and 39.

How many memory cards does Johnny-Gadget have in his pockets?

Answer 1

The number is not a multiple of 5. Due to (1) only 1,5,10,15 are possible, which violate (2). Then it is also not a multiple of 10. So the number is between 30 and 39. Then it has to be divisible by 8. So Johnny has 32 cards in his pocket.

Answer 2

He has

32 memory cards in his pocket

---

Let M be the number of memory cards:

If M is between 1-19 then it could be any number, however

- All numbers 1-19, excluding 5, 10 and 15 for now, are either not multiple of 8 or not multiples of 10 which means it cant be them.

- If it is 5, then it is not a multiple of 8 or 10, which is a contradiction.
- If it is 10, then it is not a multiple of 8 which is a contradiction.
- If it is 15, it is not a multiple of 8 or 10 which is a contradiction.

So we know it's not between 1-19, as all numbers are either not a multiple of 8 or 10.

If M is between 20-29 then it is a not a multiple of 8. Which means that is could be 20, 21, 22, 23, 25, 26, 27, 28 or 29.

- It can't be 20 or 25 as they're a multiple of 5, but not between 1-19.
- It can't be 21, 22, 23, 226, 27, 28 or 29 because they're not multiples of 10 but aren't in 30-39.

So we know it's not between 20-29.

If M is between 30-39, then it's not a multiple of 10. Which means it could be 31-39.

- It can't be 35 because that's a multiple of 5.
- It can't be 31, 33, 34, 35, 36, 37, 38 or 39 as these are not multiples of 8 but aren't in 20-29.

Therefore the only number it can be is

32

as all numbers greater than 39 are

either a multiple of 5, or not a multiple of 8 or 10, which will lead to a contradiction.

Answer 3

Summarizing the hints:

1. If a multiple of 5, must be 1-19, so one of (5, 10, 15)
2. If not a multiple of 8, must be 20-29, so one of (20, 21, 22, 23, 25, 26, 27, 28, 29)
3. If not a multiple of 10, must be 30-39, so one of (31, 32, 33, 34, 35, 36, 37, 38, 39)

What are the options?

Looking at hints 2 and 3, if the number is not a multiple of 8 it must be between 20-29, and if it is not a multiple of 10 it must be between 30-39. These two ranges do not overlap, so if it were not a multiple of 8 or 10 the number would have to be in both ranges. Take 29 for example. That's not a multiple of 8 so hint 2 says it must be between 20-29. But hint 3 says it must be between 30-39.

Therefore the number is either a multiple of 8, a multiple of 10, or both.

Is it a multiple of 10?

Every multiple of 10 is also a multiple of 5. According to hint 1, if it is a multiple of 5 (and therefore 10) it must be in the range 1-19. The only multiple of 10 in that range is 10. But that is not a multiple of 8, so hint 2 tells us the number has to be in the range 20-29 if it is not a multiple of 8.

Therefore the number is not a multiple of 10

So

We've shown that it must be a multiple of 8, 10, or 8 and 10. But we've shown it cannot be a multiple of 10, so it must be a multiple of 8. Because it isn't a multiple of 10, hint 3 tells us it must be in the range 30-39. The only multiple of 8 in that range is 32.

Therefore the answer is 32.

Answer 4

I'm just sharing my approach here because I think it's a bit more algorithmic than others. So we have 3 questions to ask a potential solution (divisible by 5, 8, 10?) - if we make zero assumptions about what is possible then the set of answers can be 8 possible sets. The truth table below sets out the possible answers, and then from there we can infer three other things about each answer - the range that is is in.

+---+----+----+-----+------+-------+-------+
| x | %5 | %8 | %10 | 1-19 | 20-29 | 30-39 |
+---+----+----+-----+------+-------+-------+
| 1 |  0 |  0 |   0 |      |     1 |     1 |
| 2 |  0 |  0 |   1 |      |       |       |
| 3 |  0 |  1 |   0 |      |       |     1 |
| 4 |  0 |  1 |   1 |      |       |       |
| 5 |  1 |  0 |   0 |    1 |     1 |     1 |
| 6 |  1 |  0 |   1 |    1 |     1 |       |
| 7 |  1 |  1 |   0 |    1 |       |     1 |
| 8 |  1 |  1 |   1 |    1 |       |       |
+---+----+----+-----+------+-------+-------+

So now we just need to use maths rules to eliminate rows that have contradictions:

No number can be in more than one of these ranges, so rows 1, 5, 6, 7 get chucked out.
No number can be a multiple of 10 but not of 5, so goodbye rows 2 and 4.

That leaves:

Row 3, the set of numbers divisible by 8, but not 5 or 10, between 30 and 39.
Row 8, the set of numbers divisible by 8, 5, and 10, between 1 and 19.

And now our answer:

The row 2 set doesn't have any numbers in it, and there's only one in the row 1 set, which is 32, our answer.

This method was probably a bit overkill for this problem, but I definitely found it a lot easier and less error-prone than attempting others.