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 Odd Digits 2 Differ Adjacently (Posted on 2008-09-22)
Consider all possible 1000 digit positive base 10 integers all of whose digits are odd. For how many of these integers, do each pair of adjacent digits differ precisely by 2 ?

Note: Try to solve this problem analytically, although computer program/ spreadsheet solutions are welcome.

 See The Solution Submitted by K Sengupta Rating: 4.0000 (1 votes)

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 Solution Comment 5 of 5 |
`According to given condition,1) If a number is 3 or 7, then its alternate number isalso 3 or 7.No. of 3 digit numbers formed with starting digiteither 3 or 7 = 3 ( {353,313,357} or {753,757,797}Let 1st 3 or 7 can occur in 1st place or 2nd place,so next 2 digits can be selected in 3 ways, from thereusing the 2nd appended digit (which is 3 or 7 again)next 2 digits can be selected in 3 waysSo, 4 ways of choosing either 3 or 7 and 1st place or 2ndplace.if its the second place 1st digit can be selected in 2 waysif its the 1st place, last digit can be selected in 2 waysSo, total 4*2=8 ways here.from there every 2 digits can be selected in 3 ways using last digit again.So, there are 3^(1000-2)/2 ways of choosing the numbersFinally 8*3^499 numbers satisfy the given criteria.`

 Posted by Praneeth on 2008-10-07 07:17:48

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