How long will it take to topple 1 million dominoes?

If you were able to arrange 1 million dominoes so that each falling tile topples exactly two tiles, and assuming 1 second of time required for each of these double topples, how much time would be required to topple all the tiles?|||The sequence with which the dominoes topple is the following:





1. 1 so far 1


2. 2 so far 3


3. 4 so far 7


4. 8 so far 15


5. 16 so far 31


6. 32 so far 63





...and so on. Note that the number of dominoes toppled at the N-th second is 2^N, and that the number of dominoes toppled just before that is always 2^N - 1. Therefore, the total number of dominoes after N seconds is always 2^N + 2^N - 1, that is:





T = 2^N + 2^N - 1





Or:





T = 2(2^N) - 1


T = 2^(N+1) - 1





Now, for this number to be greater than 1,000,000, the following inequality is required:





2^(N+1) - 1 %26gt;= 1000000





Solving he inequality:





2^(N+1) %26gt;= 1000001


log(2^(N+1)) %26gt;= log(1000001)


(N+1)*log(2) %26gt;= log(1000001)


N+1 %26gt;= 19,93


N %26gt;= 18,93





So, the correct answer the number of toppled dominoes surpasses 1,000,000 after 19 seconds.|||1 + (from i = 1 to n)危 2i = 1,000,000





(from i = 1 to n)危 2i = 999,999





2 (from i = 1 to n)危 i = 999,999





n(n+1) = 999,999





n^2 + n - 999,999 = 0





n = 999.499 ~ round up -%26gt; 1,000


1,000 seconds.|||1 x 10^6 / 2 = 500000 seconds /60 = 8333.33 minutes / 60 = 138. 8 hours / 24 = 5.7 days. holy|||A long time