Posted by Sky on Friday, August 31, 2012 at 7:29pm.
I'm not Bob, and he may have further input, but it seems to me you just take what information they give you, and step by step apply it.
They said:
You can only use the digits 0-9. I need an odd number that is a multiple of five with no repeated digits. Half the digits are odd and the other half is even. The largest digit in the number is in the tens place (but it is not the largest digit). The digit in the hundreds place is half the digit in the tens place. The sum of the digits is greater than 20.
you know the number is odd, and a multiple of five, so it must end in 5. Explain how you know this.
The largest digit is 9, but the number in the 10s place is not 9, so it must be 8 or less.
Since the digit in the hundreds place is half the tens' place, the tens digit must be even, so it's 0,2,4,6,8. But, it can't be 0,2,or 4, so it must be 6 or 8.
Why, because the sum of the digits is 20, and if the largest digit is 6, then adding 0-6 gives 21.
Now, we don't know how many digits there are in the number, but there must be at least 6, if the largest is 6. That would give us the choices
xxx365 if there is no 0, or
xxxx365 if there is a 0 in the mix
Since half the digits are odd, there must be an even number of digits. so there is no 0.
xxx365 is the only pattern. Using that, we can have only one more odd digit, so fill in the rest as you will, using 1,2,4 anywhere you like.
Now, if the 10's digit is 8, then we could have
xxx485 or xxxxx485
The sum must be greater than 20, but any of the remaining digits will work.
So, using 0,1,2,3,6,7 fill in the other digits, making sure that 0 is not the first digit, there are the same number of evens and odds, and there are no repeats.
Bob's 123485 is just one possible solution. If they wanted the smallest possible number, then I'd have said 103485. The largest would be 76321485.
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