Firstly, I began with the simplest of options in this challenge:
1,1,1,1,1,1,1,1,1,1,1,1
AND
2,2,2,2,2,2
Then, using the all 1’s sequence, I replaced 1’s with 2’s in all the places possible, and added them to the list of possibilities in the challenge, E.G:
• 2,1,1,1,1,1,1,1,1,1,1
• 1,2,1,1,1,1,1,1,1,1,1
• 1,1,2,1,1,1,1,1,1,1,1
• 1,1,1,2,1,1,1,1,1,1,1
ETC.
(This came to 11 possibilities in total)
Next, I began to continue with the same thing; gradually combining more and more 2’s and added the amount of this possibility to the total. I.E. for this group, I added two 2’s to the possibilities:
• 2,2,1,1,1,1,1,1,1,1
• 1,2,2,1,1,1,1,1,1,1
• 1,1,2,2,1,1,1,1,1,1
• 1,1,1,2,2,1,1,1,1,1
ETC.
(This came to 43 possibilities in total)
Following the same routine of adding one more 2 each time, I used three 2’s in this sequence and added them to the list:
• 2,2,2,1,1,1,1,1,1
• 1,2,2,2,1,1,1,1,1
• 1,1,2,2,2,1,1,1,1
• 1,1,1,2,2,2,1,1,1
ETC.
(This came to a total of 84 possibilities)
Striving on, the next amount of 2’s was consequently 4. Therefore there was an even split of both 1’s and 2’s; which meant I only had to find out the sequences beginning with 2 and then double it:
• 2,2,2,1,2,1,1,1
• 2,2,2,1,1,1,1,2
• 2,2,2,1,1,1,2,1
• 2,2,2,1,1,2,1,1
ETC.
(This came to a total of 36 possibilities)
To complete the challenge, I added five 2’s to the sequence because six 2’s meant the whole row was 2’s:
• 2,2,2,2,2,1,1
• 2,2,2,2,1,2,1
• 2,2,2,1,2,2,1
• 2,2,1,2,2,2,1
ETC.
(This came to a total of 21 possibilities)
Now all I had to do was add all of the possibilities together to therefore generate the grand total:
21
36
84
43
11
2+
=197