So Many Possibilities!

Probability Level 3

Let $S_{1}=[1,2,3]$ . For any $S_{n}=[a,b,c]$ , define $S_{n+1}=[b+c,b,c]$ , that is, exactly one element from $S_{n}$ is replaced with the sum of the other two elements. For how many values of $n$ does $S_{n}=[1234,2345,3456]$ ?

For example, if $S_{133}=[1234,2345,3456]$ and $S_{583}=[1234,2345,3456]$ , there would be two values of $n$ that would contain the set $[1234,2345,3456]$ , and your answer would be 2.

The answer is 0.

1 solution

Linus Setiabrata
Aug 27, 2014

Note that $S_{1}$ has two odd numbers (1 and 3) and one even number (2). We then have two cases:

Case 1.

We replace the even number with the sum of the other two odd numbers.

Then, the replacement number would stay even.

Case 2.

We replace one of the odd numbers with the sum of the other two numbers.

Then, the replacement would be the sum of an even and odd number, which yields an odd number.

Therefore, no matter what numbers we replace, we'll always have two odd numbers and one even number.

It is easy to see that the set $[1234,2345,3456]$ has two even numbers, so it is impossible to obtain.

Hence, the number of possible values for $n$ that would satisfy $S_{n}=[1234,2345,3456]$ is $\boxed{0}$

So Many Possibilities!

The answer is 0.

1 solution

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