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# 10-digit Number

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10-digit Number - Back to the Number Puzzles

* Find a 10-digit number, where the first figure defines the count of zeros in this number, the second figure the count of numeral 1 in this number etc. At the end the tenth numeral character expresses the count of the numeral 9 in this number.

* Find a 10-digit number, where the first numeral character expresses the count of numeral 1 in this number, ..., the tenth numeral the count of zeros in this number.

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10-Digit Number - solution

* Sum of all numerals must be ten because each numeral stands for the count of other numerals and because this number shall have ten numerals. Beginning to choose reasonable numerals for the first figure you can come across the correct number: 6210001000.

* 2100010006.

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• 3 months later...

I don't understand the question very well. What is a 'count of zeros,' there should only be one zero in this number right?

You can tell me if I've got it right...

We need a ten digit number: XXX-XXX-XXXX

And the value of the first digit gives the placement of the number 0, for example: 5XX-X0X-XXXX.

Is that what you're looking for?

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Oh, I think I understand, you're looking for these numbers?

6210001000

2100010006

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Yes, the part I don't understand is why it was two separate questions. The value of the zeros doesn't change when you change the digit where you count them. That's why I originally thought you were looking for the position of each, because you move the position in the second half of the question.

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feel free to reword the puzzle to make it more comprehensible

9000000000

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9000000000

so there's nine 0s and zero 9s?

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6210001000.

Follow the process of substitution starting with 9000000000.

9000000000 --> 8000000010 --> .... --> 6210001000.

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• 1 month later...

Which part of the question said all 10 digit must add up to equal to 10?

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• 2 weeks later...
Which part of the question said all 10 digit must add up to equal to 10?

Logic says that. If each place value (billions' place through ones' place) is a count of how many times a numeral (0 through 9) appears in the 10 digit number then the sum of the digits will equal 10 because because there are only 10 numerals to be counted.

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• 1 month later...

Can this be a solution?

9000000001

1000000008

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Can this be a solution?

9000000001

1000000008

no

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• 3 weeks later...

ans are

6210001000

&

2100010006

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• 5 weeks later...

i dont understand why it has to add up to anything

9,000,000,000

this fits as a solution to the first half anyway

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9000000000 cannot be an answer because the number of 9's in the full ten digit number is 1 not 0. The number would be 9000000001. This cannot be because now there are only eight 0's.

Solution (color indicates wrong value based on the count of respective numbers):

9000000000

9000000001

8100000001

8100000010*

*This is an infinite loop. If there is one 8 then there is one 1 in the 9th digit for the number of 8's and one 1 in the 2nd digit for the number of 1's. However there are actually two 1's in the ten digit number. If this is changed to 2 then there is only one 1 and this continues forever... Therefore it is impossible for a single 1 to exist in the ten digit number.

The second digit must be 0, 2 or greater. If the number of zeros previously is an indication, there will be more than one 0, creating a 1 somewhere in the digit. So if the second digit is 2, the third digit will be 1 and another 1 will be placed in the appropriate spot for the number of 0's left.

6210001000

The second part is actually easier since the numbers have already been figured out. If there are six 0's, two 1's and one 2 then:

2100010006

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• 2 weeks later...
i dont understand why it has to add up to anything

9,000,000,000

this fits as a solution to the first half anyway

You are not understanding that the digits in the number have to represent the count of digits in the number itself. 9,000,000,000 shows that you have nine zeros, which is true, but the final zero also indicates that you have zero nines, which is false.

This might be easier to understand with an example that is at once harder than the problem and potentially easier to understand. Consider the following sentence:

"This sentence contains XX words, XX syllables, and XX letters."

How do you solve it? As you put in different words, the counts of the various parts change. Counting up the words not including the XXs, we get:

"This sentence contains seven words, XX syllables, and XX letters."

But we added the word "seven", so the sentence now contains eight words! We need to change it to:

"This sentence contains eight words, XX syllables, and XX letters."

We can see that when we replace the other XXs with words, we will in fact have a total of ten words:

"This sentence contains ten words, XX syllables, and XX letters."

Our sentence now contains thirteen syllables, not including the XXs:

"This sentence contains ten words, thirteen syllables, and XX letters."

But since "thirteen" itself contains two more syllables, we actually have fifteen syllables! Fortunately, "fifteen" also has two syllables, so we can say:

"This sentence contains ten words, fifteen syllables, and XX letters."

Counting up the letters so far, we have 4 + 8 + 8 + 3 + 5 + 7 + 9 + 3 + 7 = 54 letters:

"This sentence contains ten words, fifteen syllables, and fifty-four letters."

But this is now wrong. "Fifty-four" itself has nine more letters, for a total of 63 letters. In addition, the sentence no longer has fifteen syllables; it now has 18 syllables:

"This sentence contains ten words, eighteen syllables, and sixty-three letters."

But this is still wrong, as we knew it would be. Both "eighteen" and "sixty-three" have one letter more than the words they replaced ("fifteen" and "fifty-four", respectively). A quick count verifies that there are now 65 letters:

"This sentence contains ten words, eighteen syllables, and sixty-five letters."

But since "sixty-five" has one letter fewer than "sixty-three", our actual letter count is 64 -- which, thankfully, has the same written letter count as 65:

"This sentence contains ten words, eighteen syllables, and sixty-four letters."

This sentence is now perfectly self-describing. In the same sense, 621001000 is perfectly self-describing if you understand the first digit to represent the number of 0s in the number, the second digit to represent the number of 1s in the number, and so on to the tenth digit, which represents the number of 9s in the number.

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• 2 months later...

This one is pretty easy... work backwards if you name the number of zeros in the first position, then you will need to have one of that digit ... so that leaves 6 spots left for the zeros. Therefore the first number is 6 2 1 0 0 0 0 1 0 0 -six zeros, 2 ones, 1 two and 1 seven. (note - I used trial and error to back into 6. I actually first had 7, but then realized I had two ones to have account for that caused using a two)

Flipping it its the same combo of numbers just a different order... 2 1 0 0 0 1 0 0 0 6

Edited by RiddleRookie
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• 1 month later...

good point

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• 1 month later...

Answer to first question: 6,210,010,000

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• 1 month later...
10-digit Number - Back to the Number Puzzles

* Find a 10-digit number, where the first figure defines the count of zeros in this number, the second figure the count of numeral 1 in this number etc. At the end the tenth numeral character expresses the count of the numeral 9 in this number.

* Find a 10-digit number, where the first numeral character expresses the count of numeral 1 in this number, ..., the tenth numeral the count of zeros in this number.

10-Digit Number - solution

* Sum of all numerals must be ten because each numeral stands for the count of other numerals and because this number shall have ten numerals. Beginning to choose reasonable numerals for the first figure you can come across the correct number: 6210001000.

* 2100010006.

I got the same answer, but is it unique?

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• 3 months later...

Would the answer to the first half of the question be....

1234543210 ?

1 is the number of 0's

2 is the number of 1's

0 is the number of 9's

in the 10 digit number

or maybe 1233233210

Edited by spentonbargains
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• 3 weeks later...
Would the answer to the first half of the question be....

1234543210 ?

1 is the number of 0's

2 is the number of 1's

0 is the number of 9's

in the 10 digit number

or maybe 1233233210

No.

1 is the # of 0's, 2 is the # of 1's, and 0 is the # of 9's, but 3 is not the # of 2's, 4 is not the # of 3's, etc.

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