mersenneforum.org 10 digit number puzzle
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 2014-11-08, 11:17 #1 MattcAnderson     "Matthew Anderson" Dec 2010 Oregon, USA 24·32·5 Posts 10 digit number puzzle Hi fun math people, I got this one from www.mathisfun.com :) I'm sure it will be too easy for some of you. Find a 10-digit number where the first digit is how many zeros in the number, the second digit is how many 1s in the number etc. until the tenth digit which is how many 9s in the number. Best of luck, Matt
 2014-11-08, 14:09 #2 Luis     Oct 2014 Bari, Italy 3·13 Posts 6210001000? It should be nice to find the general case. Last fiddled with by Luis on 2014-11-08 at 14:30
 2014-11-08, 14:17 #3 science_man_88     "Forget I exist" Jul 2009 Dumbassville 203008 Posts 9,000,000,000 8,100,000,000 probably more than these I'm just not able to see them easily.
 2014-11-08, 14:27 #4 retina Undefined     "The unspeakable one" Jun 2006 My evil lair 6,143 Posts This is an old puzzle. And since the first two posters haven't yet found a result I'll not give away a solution edit: I see one poster has since edited and given a solution for base 10. But there is a simple function that can generate such numbers for any arbitrary integer base except 2, 3, 6 and 7. I think it would be more interesting to try and find the general function first and then get the answer for the OPs question in base 10. Last fiddled with by retina on 2014-11-08 at 14:28
2014-11-08, 14:29   #5
Luis

Oct 2014
Bari, Italy

3×13 Posts

Quote:
 Originally Posted by science_man_88 9,000,000,000 8,100,000,000 probably more than these I'm just not able to see them easily.
They are wrong. ;) Check the tenth digit in the first one and the nineth digit in the second one.

2014-11-08, 15:45   #6
Luis

Oct 2014
Bari, Italy

3·13 Posts

Quote:
 Originally Posted by retina This is an old puzzle. And since the first two posters haven't yet found a result I'll not give away a solution edit: I see one poster has since edited and given a solution for base 10.
Does it mean that base 10 has more than one solution?

Quote:
 But there is a simple function that can generate such numbers for any arbitrary integer base except 2, 3, 6 and 7. I think it would be more interesting to try and find the general function first and then get the answer for the OPs question in base 10.
I found a function that generates one solution for base >= 7.

digit_i, i=0, ... base-1

digit_0 = base-4 (it's not a number if base > 10)
digit_1 = 2
digit_2 = 1
digit_(base-4) = 1
digit_i = 0 otherwise

A base 4 solution is 1210, but I can't find solutions for base 5...

Last fiddled with by Luis on 2014-11-08 at 15:48

2014-11-08, 16:03   #7
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

838410 Posts

Quote:
 Originally Posted by Luis They are wrong. ;) Check the tenth digit in the first one and the nineth digit in the second one.
9,210,000,001 you mean ?
edit: doh forgot to change the first digit to 6 instead fo 9 so change one of the 1's, so 6,210,001,000

Last fiddled with by science_man_88 on 2014-11-08 at 16:07

2014-11-08, 16:54   #8
Luis

Oct 2014
Bari, Italy

3·13 Posts

Quote:
 Originally Posted by science_man_88 9,210,000,001 you mean ? edit: doh forgot to change the first digit to 6 instead fo 9 so change one of the 1's, so 6,210,001,000
For instance:
9,000,000,000
wrong because you have 1 'nine', so the tenth digit should be 'one'.
9,000,000,001
wrong because there are not 9 'zero's, but only 8.
8,000,000,010
wrong, because you have 1 'one', so the second digit should be 'one'. If the second digit is 'one', you have 2 'one's.
And so on...

6,210,001,000 is the only solution I found for base 10. I don't know if there are another ones.

2014-11-08, 19:21   #9
TheMawn

May 2013
East. Always East.

11×157 Posts

I wrote a fairly neat little Excel Spreadsheet for this one.

It's an iterative process where each cell represents one digit and the formula for the cell is the count of each digit.

It started with 9,000,000,000 and iterated to 8,000,000,001 and eventually to 6,210,001,000.

I next added a bit of code which changed the starting values to random numbers between 0 and 9 if it reached 6,210,010,000. The intention is to "reset" the game to a new start point automatically. The solution should "freeze" at another valid solution if it finds a new one since the "count" of digits will not change from iteration to iteration, but it will also not reset since it didn't reach 6,210,001,000.

I have attached it for your enjoyment. You can notice the iterative process and then the change when it reaches the currently known solution. Excel 2010. It "SHOULD" default to manual calculation mode AND allowing iterative calculations, which means you must simply hold down the F9 button (unless you've changed your manual calculation button).
Attached Files
 Puzzle.zip (6.2 KB, 77 views)

Last fiddled with by TheMawn on 2014-11-08 at 19:21

 2014-11-08, 19:34 #10 TheMawn     May 2013 East. Always East. 11×157 Posts For 9 digits: 521,001,000 For 8 digits: 42,101,000 For 7 digits: 3,211,000 For 4 digits: 1,210 There's a clear pattern in going from 9 digits down but once the pattern is impossible to fit there appear to be no solutions. I can't find any for 6 digits or 5. Four digits appears to be a special case in this regard.
 2014-11-08, 19:39 #11 TheMawn     May 2013 East. Always East. 172710 Posts And sure enough: B21,000,000,001,000 A2,100,000,001,000 9,210,000,001,000 821,000,001,000 72,100,001,000 Essentially, there's as many 0's as you want, a first digit indicating that number of zeros, and then a 1 for that digit. Then, you trigger an extra 1 for the number of 1's, giving you a 2 for the number of 1's, giving you a 1 for the number of 2's (which conveniently replaces the 1 for the number of 1's you had earlier. Last fiddled with by TheMawn on 2014-11-08 at 19:44

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