Friday Puzzle -- a wee bit late

Re: Friday Puzzle -- a wee bit late

Larry Barrett

Lots of possible answers to this, depending on how you read it.

Answer to one way is 32.

Hmm

Alex Y

I'm not seeing the interpretation that gets you to 32.

Anyway, my intent was to get the digit count of each person, then take the product across the earth's population. E.g., for my immediate family, there are four of us with 20 digits apiece, so our product would be 160,000.

Re: Hmm

Larry Barrett

Population = 6B = 6(10*9)

Count of each persons fingers = 8, product for all persons = 8x6(10*9)

Count of each persons thumbs = 2, product for all persons = 2x6(10*9)

Count of each persons toes = 10, product for all persons = 10x6(10*9)

Product of products = 48x12x60(10*27) = 34.56(10*30)

Number of digits in product = 32.

...................

What you intended is of course much larger, 20*6B = (2*6B)(10*6B).

Problem becomes how to estimate number of digits in 2*6B.

2*10 = 1024 ~= 10*3. Good enough for talking about computer memory a few decades ago, but probably not good enough for estimating 2*6B.

Still thinking about this.

Hint

Alex Y

The solution requires an "Aha!", not calculation.

Clarification

Alex Y

The two different uses of the word "digit" in the statement of the problem is not significant. One refers to "fingers, thumbs, and toes" while the other refers to numerals, with no double entendre intended.

10

Re: Hint

Larry Barrett

This is not exactly an aha, so probably not correct.

Looking at the product (20)(20)(20)... for the first several values, and then looking at the number of digits in those products:

N Product #digits

1 20 2

2 400 3

3 8000 4

4 160000 6

5 3200000 7

...

It looks like the number of digits in the Nth product = (N+1) plus some extra digits every so often.

When I carried this out for N up to 32, it looks like the extra digits = 10.

When N = 32, the number of digits in the product = 32+1 + 10 = 43

So for 6B people the number of digits in the product = (6B+1) + 10(6B/32) = (6B+1) + 1,187,500,000.

11 counting hands and feet

Alex Y

Neither is the number of digits I have in mind for the answer. But out of curiosity, how did you come up with that?

Alex Y

Sorry, no. Not "aha" enough;0

Another hint

Alex Y

This hint is somewhat circular, but still:

Think of the difference between an arithmetic average and a geometric average.

The arithmetic average of number of digits per person is darned close to 20 -- probably 19.9 followed by a few additional 9's. So using 20* 6B as an estimate of the sum of the digits of all individuals is very good. But to use n^6B to estimate the product of all individuals' digit counts, you need to set n equal to the geometric average of the number of digits per person, the surprising value of which will lead to the puzzle answer (or vice-versa, because of the referenced circularity of this hint)

Final hint/answer

Alex Y

Final hint: Do you think that among the 6b people, there is at least one quadruple amputee?

Answer below

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Answer: 1

Assuming there is such a person (of which I have no doubt) or a person who due to birth defects has no digits, it doesn't matter how large the number is that you get by multiplying all of the 20s together; when you multiply by this person's number of digits, you get the final answer of zero, which of course is expressed using only one digit.

And I find it interesting that while I usually think of geometric mean and arithmetic mean as being pretty close, the existence of a zero in the group being averaged certainly changes that. For digits per person, the arithmetic mean is 19.99.. while the geometric mean is zero.

After your first hint I considered

Gary Smyth

the problem was a "trick" question. My answer was still wrong but the wording seems ambiguous. My thinking was that if you have a population of 6 billion it seems to me that I can statistically assume that as humans most have 20 digits per person if you count fingers and toes (1 finger = 1 digit, 2 fingers = 2 digits...) you come up with 120,000,000,000 digits. That number, 120 billion, has 11 digits, ooops ...12 unless you counted in British billions then it would be 15.

Re: After your first hint I considered

Alex Y

My answer was still wrong but the wording seems ambiguous.

Yeah, and I'm sorry about that. I struggled with the wording. Maybe I should have used the example of the product for my immediate family to make clearer what I meant.

if you count fingers and toes (1 finger = 1 digit, 2 fingers = 2 digits...) you come up with 120,000,000,000 digits. That number, 120 billion, has 11 digits, ooops ...12

But that is the sum across the population of the number of digits. I was looking for the product.

unless you counted in British billions then it would be 15

LOL. As if we don't have enough problem with overpopulation as it is!

Thanks for the lighthearted reply.

Re: Final hint/answer

Larry Barrett

A good "aha". I did think of people with less than 20 digits, as well as more (thinking of recent news about child in Burma with 12 fingers and 14 toes), but never got to zero.