Calvin and Susan play a Number-Game #2
Calvin writes a number with 2187 digits on the blackboard, each digit being either a 1 or 2. Susan creates a new number from Calvin's number by reading his number from left to right and wherever she sees a 1, replacing that by 112, and wherever she sees a 2, replacing that by 111.
For example : If Calvin's number begins , Susan's number would begin .
After Susan finishes writing her number, she notices that the left-most 2187 digits in her number is exactly same as that of Calvin's number. How many times do five s occur consecutively in Calvin's number?
Also try it's sister problem, an easy version: Calvin and Susan play a Number-Game #1
Image Credit: Dreamstime
The answer is 182.
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Set S: Numbers Susan writes; Set C: Numbers Calvin writes
Since 1 translates to 112 and 2 translates to 111, Set S has 3 times the number of elements in Set C. Set S is a translation/mapping of Set C. Thus, first 2187 numbers of Set S are a translation of first 729 numbers of Set C. But, the first 2187 numbers of both Sets are equal! Thus, the 2187 numbers of Set C is a translation of its first 729 numbers. Similarly, the first 729 numbers of Set C is a translation of its first 243 numbers. (Same logic: First 729 numbers of Set C= first 729 numbers of Set S. But, the first 729 numbers of Set S= translation of first 243 numbers of Set C. Therefore, the first 729 numbers of Set C = translation of first 243 numbers of Set C)
From now on, the set under consideration is always Set C unless specified otherwise. The first number is 1, since both sets have to have the same first number. The 3rd term is 2, since 1 would translate into 112. The 9th term is 1, since the 3rd term would translate into 111. The 27th term is 1. . . The 729th term is 1. The 2187th term is 2.
It is also to be noted that 2 or more consecutive 2s cannot appear. This is because every part of Set C is a translation of some part before it, and a translation of any of 11, 12, 21 and 22 would not give 2 consecutive 2s.
To find: No. of 5 consecutive 1s. Translation of 11 does not give 5 consecutive 1s. 22 is not present in Set C. Also, every 12 and 21 is actually a 121. (2 consecutive 2s not possible) Translation of a 121 gives 5 consecutive 1s.
To find: No. of 121s in first 729 terms. Since 1st term is a 1, To find: No. 21s in first 729 terms. Also the 729th number is a 1.
Thus the problem reduces to finding the number of 2s in the first 729 terms. Let N(2,729) denote the required number. N(2,729) = N(1,243) (only 1 can translate to give a 2) where N(1,243) is the number of 1s in first 243 terms. N(1,243) = 243 - N(2,243) = 243 - N(1,81) = 243 - 81 + N(2,81) = 162 + N(1,27) = 162 + 27 - N(2,27) = 189 - N(1,9) = 180 - 9 + N(2,9) = 180 + N(1,3) = 180 + 3 - N(1,1) = 182 Since the number of 1s in the first 1 term(s) is 1.
The answer is 182.