Friday Puzzler--10-digit number

Re: Friday Puzzler--10-digit number

Henry Higginbotham

Some at random that seem to work:

8 100 000 000

6 211 000 000

5 211 300 000

2 511 111 880

7 000 000 370

2 025 333 670

. . . unless I misread the conditions, even after having coffee today. I kind of suspect the last digit will always be zero, but that notion is purely empirical. As for how many, seems like "more than several," but my math background is too weak to go much further.

Re: Friday Puzzler--10-digit number

Larry Barrett

Simplify the problem a little to read 'is there a one digit number where the first digit is the number of 0s?'

If you say that the answer is 0, then there is one 0, so the first digit must therefore be 1. But if it is 1, then there are no 0s.

How about 'find a two digit number where the first digit is the number of 0s, the second digit is the number of 1s'.

The possible answers are 00, 01, 10, 11.

The problem with 00 is that there are two 0s, so this must be changed to 20, but now there is only one 0.

The problem with 01 is that there is one zero, so this must be changed to 11, and now there are zero 0s and two 1s.

The problem with 10 is that there is one 1, so this must be changed to 11, but now there are zero 0s, etc.

Similar problems with 11.

I think there is no solution to the original problem.

Ah, I missed the ellipsis

Henry Higginbotham

I thought that sounded too easy.

Re: Friday Puzzler--10-digit number

Alex Y

A little too quick to generalize from the cases of length 1 and 2. Consider that 1210 works for the four-digit case. Not saying necessarily that there is a solution for ten digits.

Re: Friday Puzzler--10-digit number

Larry Barrett

and 21200 works for a 5 digit number.

Observation so far is that the sum of the digits must equal the number of digits.

More to it than that.

Re: Friday Puzzler--10-digit number

Larry Barrett

Somber news from France tonight.

Some solutions (I think). I don't understand the pattern so far.

--Digits- 0 1 2 3 4 5 6 7 8 9

1 digit---No solution

2 digit---No solution

3 digit---No solution

4 digit---1 2 1 0 (this is Alex's solution)

5 digit---2 1 1 0 0

6 digit---I have not found a solution

7 digit---3 2 1 1 0 0 0

8 digit---4 2 1 0 1 0 0 0

9 digit---5 2 1 0 0 1 0 0 0

10 digit--6 2 1 0 0 0 1 0 0 0

Got one! Others?

Alex Y

Very good!

BTW, typo in your list for 5. S/b 21200, as you said in previous post. And there is no solution for 6.

Now, what other solutions exist for 10, or can you show that there are none?