It is also possible to do this combinatorially.
If allowed digits are 1,2,3, and we have six slots to use, then there are a
total of 3^6 = 729 arrangements.
So we just need to count all arrangements that have 3 or more consecutive
digits that are the same.
Consider the slots as such
_ _ _ _ _ _
Use | (pipe) to denote the last slot that is in a 3-series (3 or more
consecutive digits). Important point is that we are choosing the LAST
slot that is part of a 3-or-more series.
e.g.
_ _ _ | _ _ _
can be
1 1 1 | _ _ _
2 2 2 | _ _ _
3 3 3 | _ _ _
where the right hand of | needs to be counted.
There are 3 ways of choosing the LHS. RHS depends on whether index 4 and index
5 slots contain the same digit.
If they do then there are
3 * 2 * 2 ways.
If not then there are
3* 2 * 2 * 3 ways.
Next consider
_ _ _ _ | _ _
The first item can be any. Use 3 to denote that it canbe any three of the
digits.
3 _ _ _ | _ _
Using the previous argument we have
3 * 3 * 2 * 3 ways.
Next,
_ _ _ _ _ | _
We have 3 * 3 *3 * 2 ways.
Lastly
_ _ _ _ _ _|
We have 3 ^ 4.
Total is (3^4) + (3*3*3*2) + (3*3*2*3) + (3*2*2*3) + (3*2*2)
Subtract from the total number of possible arrangements:
729 - (3^4) + (3*3*3*2) + (3*3*2*3) + (3*2*2*3) + (3*2*2)
492
--------------------------------------------
On Mon, 4/9/18, 'Jon Hough' via Programming <programm...@jsoftware.com> wrote:
Subject: Re: [Jprogramming] Quora Problem
To: programm...@jsoftware.com
Date: Monday, April 9, 2018, 3:57 PM
Not the fastest method...
f=:
-.@:(1&e.)@:(#@:~."1)@(3&(]\))`0:@.(1&e.@:('0456789'&e.))@:":
D=: 1e5 + i. 1e6 - 1e5
+/ f"0 D
492
--------------------------------------------
On Mon, 4/9/18, Skip Cave <s...@caveconsulting.com>
wrote:
Subject: [Jprogramming] Quora Problem
To: "programm...@jsoftware.com"
<programm...@jsoftware.com>
Date: Monday, April 9, 2018, 2:51 PM
Here's a fun math challenge on Quora:
Find the number of 6-digit numbers
made
up of the digits 1, 2, and 3 which
have no digit recurring three or more
times, consecutively?
The link to my answer on Quora is: https://goo.gl/BzBDQe
Skip Cave
Cave Consulting LLC
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