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Topic: What a number! (Read 479 times) |
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pcbouhid
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What a number!
« on: Dec 12th, 2005, 4:54pm » |
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Find a positive integer (base 10) which has the following four properties:
(a) Each of the 10 digits (0 to 9) appears exactly once in the number.
(b) For each pair of digits whose sum is 9, the number of other digits positioned strictly between the pair is equal to the smaller digit of the pair. Explaining better: between 2 an 7 (that sums to 9) there are 2 (the smallest number of the pair (2,7)) digits.
(c) The sum of each pair of digits positioned at the same distance from opposite ends of the number is a prime number.
(d) The difference between any 2 adjacent digits is greater than 1.
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| « Last Edit: Jan 4th, 2006, 8:09am by pcbouhid » |
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almaeihm
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Re: WHAT A NUMBER!
« Reply #1 on: Dec 12th, 2005, 11:32pm » |
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I'm not proud of the way I worked this problem out... it was not intellectual at all... and rather time-consuming…
I used a word file... made a table with 10 slots... grouped the numbers into five groups of two so that each group can be dragged and dropped as one entity that meets condition# 2... then I started shuffling the groups around and flipping the pairs over… and finally I got this number which meets all conditions:
5831746290
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pcbouhid
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Re: WHAT A NUMBER!
« Reply #2 on: Dec 13th, 2005, 8:55am » |
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But you got it! Read "Maquiavel".
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Sjoerd Job Postmus
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Re: WHAT A NUMBER!
« Reply #3 on: Dec 14th, 2005, 11:00am » |
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Logic dictates:
0+9 = 9 => 0 digits between 0 and 9
1+8 = 9 => 1 digit between 1 and 8
2+7 = 9 => 2, etc
3+6
4+5
Let's start with the smallest range...
*09*
If, any number is around this, it has to be 2,7; 3,6 or 4,5... If none of these match, we know that 09 is on a side... Please note that the order of the sum-digits is irrelevent, I could write 2..7 or 7..2 without any change to the solution...
Trial 1: 27
*2097*
If we want to put anything around this at all, it has to be 45
*420975*
This leaves us 2 pairs, 3,6; 1,8... we can't add those without problems. But, this would be the only way.
From this follows that 09 isn't surrounded by 27.
Trial 2: 36
3.096 OR 309.6
-- 3.096 --
Let's put the 18 around the 3
138096, we're left with 2 pairs: 27 and 45.
We can't put the 45 [widest] around it, so it has to be next to it, leaving us a wider string.
So, we do't put 18 around the 3
Let's try 45
34096.5
We're left with 18 and 27
If we put 18 there, we're left with 27 => slackspace.
So. let's try 27
3409625.7
And add 18
3409625178
Yielding us a correct number.
Adding to the table below
-- 309.6 --
We know 18 causes a failure.
Let's try 45
4.30956
27, then 18
1284730956
Yet another valid number.
3409625178
1284730956
Primeness check
3409625178: 11, 11, 1, 14, 8 FALSE
1284730956: 7, 7, 17, 4, 10 FALSE
Hrm  ...
Well, at least I tried to deduct step 1 logically... !
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