# Odometer puzzler

**URL:** <https://community.cartalk.com/t/odometer-puzzler/31867>\
**Category:** Maintenance/Repairs\
**Created:** [June 14, 2009, 1:31pm UTC](https://community.cartalk.com/t/odometer-puzzler/31867 "2009-06-14T13:31:42Z")\
**Posts on this page:** 6\
**Page:** 1

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**Author:** ![psanzone](https://avatars.discourse-cdn.com/v4/letter/p/ac8455/32.png) [@psanzone](https://community.cartalk.com/u/psanzone)\
**Post date:** [June 14, 2009, 1:31pm UTC](https://community.cartalk.com/t/odometer-puzzler/31867/1 "2009-06-14T13:31:42Z")

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I didn’t understand this week’s puzzler answer. If the number 1 appears in a six digit odometer 600,000 times (as posited in the puzzler details), then does it not appear in (1 million minus 600,000) times, or 400,00 times?

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**Author:** ![BustedKnuckles](https://sea1.discourse-cdn.com/cartalk/user_avatar/community.cartalk.com/bustedknuckles/32/25_2.png) [@BustedKnuckles](https://community.cartalk.com/u/BustedKnuckles)\
**Post date:** [June 14, 2009, 4:29pm UTC](https://community.cartalk.com/t/odometer-puzzler/31867/2 "2009-06-14T16:29:42Z")

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The question was ‘how many numbers on a 6 digit odometer do not include a 1?’ This is trickier, because the previous question was ‘how many times a 1 occurs’. In the previous example, a number like 211321 would count as three occurrences, a 1 showing up in three different positions. The 600,000 number includes a mathematical calcualtion for each digit position independantly.

The new question is ‘how many numbers do not include a 1’, and is a different set of mathematical rules. They used a statistical formula to determine the number of times a 1 showing up in ANY position.

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**Author:** ![turbonium](https://avatars.discourse-cdn.com/v4/letter/t/a3d4f5/32.png) [@turbonium](https://community.cartalk.com/u/turbonium)\
**Post date:** [June 14, 2009, 7:11pm UTC](https://community.cartalk.com/t/odometer-puzzler/31867/3 "2009-06-14T19:11:48Z")

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FYI: This thread belongs under ‘The Show’ topic.

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**Author:** ![BustedKnuckles](https://sea1.discourse-cdn.com/cartalk/user_avatar/community.cartalk.com/bustedknuckles/32/25_2.png) [@BustedKnuckles](https://community.cartalk.com/u/BustedKnuckles)\
**Post date:** [June 14, 2009, 7:39pm UTC](https://community.cartalk.com/t/odometer-puzzler/31867/4 "2009-06-14T19:39:29Z")

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Your forgetting the number of 1’s between 999,999 to 200,000 and between 99,999 to 20,000, and so on.

To do it by that method, you’d need to add it this way:

0-9 = 1 occurence  
10-19 = 10 occurences. You had that one.  
99-20 = 8 more occurences.  
100-199 = 100 occurences. You had this one, too.  
999-200 = 18 more occurences.  
And so on.

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**Author:** ![turbonium](https://avatars.discourse-cdn.com/v4/letter/t/a3d4f5/32.png) [@turbonium](https://community.cartalk.com/u/turbonium)\
**Post date:** [June 17, 2009, 2:11am UTC](https://community.cartalk.com/t/odometer-puzzler/31867/5 "2009-06-17T02:11:18Z")

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Someone else posted an even easier solution without probability. Each column can have 9 digits without a ‘1’. So the total is just 9 to the 6th power.

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**Author:** ![vtcodger2](https://avatars.discourse-cdn.com/v4/letter/v/e68b1a/32.png) [@vtcodger2](https://community.cartalk.com/u/vtcodger2)\
**Post date:** [June 17, 2009, 7:07am UTC](https://community.cartalk.com/t/odometer-puzzler/31867/6 "2009-06-17T07:07:57Z")

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\*\*\*Someone else posted an even easier solution without probability. Each column can have 9 digits without a ‘1’. So the total is just 9 to the 6th power. \*\*\*

Yep

> > > count = 0  
> > > for i in range(1000000) :  
> > > … if not “1” in str(i) : count += 1  
> > > …  
> > > print count  
> > > 531441

> > > print 9\*\*6  
> > > 531441