2020 AMC 8 Problems/Problem 22

Problem

When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.

$[asy] size(300); defaultpen(linewidth(0.8)+fontsize(13)); real r = 0.05; draw((0.9,0)--(3.5,0),EndArrow(size=7)); filldraw((4,2.5)--(7,2.5)--(7,-2.5)--(4,-2.5)--cycle,gray(0.65)); fill(circle((5.5,1.25),0.8),white); fill(circle((5.5,1.25),0.5),gray(0.65)); fill((4.3,-r)--(6.7,-r)--(6.7,-1-r)--(4.3,-1-r)--cycle,white); fill((4.3,-1.25+r)--(6.7,-1.25+r)--(6.7,-2.25+r)--(4.3,-2.25+r)--cycle,white); fill((4.6,-0.25-r)--(6.4,-0.25-r)--(6.4,-0.75-r)--(4.6,-0.75-r)--cycle,gray(0.65)); fill((4.6,-1.5+r)--(6.4,-1.5+r)--(6.4,-2+r)--(4.6,-2+r)--cycle,gray(0.65)); label("N",(0.45,0)); draw((7.5,1.25)--(11.25,1.25),EndArrow(size=7)); draw((7.5,-1.25)--(11.25,-1.25),EndArrow(size=7)); label("if N is even",(9.25,1.25),N); label("if N is odd",(9.25,-1.25),N); label("\frac N2",(12,1.25)); label("3N+1",(12.6,-1.25)); [/asy]$ For example, starting with an input of $N=7,$ the machine will output $3 \cdot 7 +1 = 22.$ Then if the output is repeatedly inserted into the machine five more times, the final output is $26.$$$7 \to 22 \to 11 \to 34 \to 17 \to 52 \to 26$$When the same $6$-step process is applied to a different starting value of $N,$ the final output is $1.$ What is the sum of all such integers $N?$$$N \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to 1$$ $\textbf{(A) }73 \qquad \textbf{(B) }74 \qquad \textbf{(C) }75 \qquad \textbf{(D) }82 \qquad \textbf{(E) }83$

Solution 1

We start with the final output of $1$ and work backwards, taking care to consider all possible inputs that could have resulted in any particular output. This produces the following set of possibilities at each stage: $$\{1\}\leftarrow\{2\}\leftarrow\{4\}\leftarrow\{1,8\}\leftarrow\{2,16\}\leftarrow\{4,5,32\}\leftarrow\{1,8,10,64\}$$ where, for example, $2$ must come from $4$ (as there is no integer $n$ satisfying $3n+1=2$), but $16$ could come from $32$ or $5$ (as $\frac{32}{2} = 3 \cdot 5 + 1 = 16$, and $32$ is even while $5$ is odd). By construction, the last set in this sequence contains all the numbers which will lead to the number $1$ at the end of the $6$-step process, and their sum is $1+8+10+64=\boxed{\textbf{(E) }83}$.

Solution 2 (variant of Solution 1)

As in Solution 1, we work backwards from $1$, this time showing the possible cases in a tree diagram:

The possible numbers are those at the "leaves" of the tree (the ends of the various branches), which are $1$, $8$, $64$, and $10$. Thus the answer is $1+8+64+10=\boxed{\textbf{(E) }83}$.

Solution 3 (algebraic)

We begin by finding the inverse of the function that the machine uses. Call the input $I$ and the output $O$. If $I$ is even, $O=\frac{I}{2}$, and if $I$ is odd, $O=3I+1$. We can therefore see that $I=2O$ when $I$ is even and $I=\frac{O-1}{3}$ when $I$ is odd. Therefore, starting with $1$, if $I$ is even, $I=2$, and if $I$ is odd, $I=0$, but the latter is not valid since $0$ is not actually odd. This means that the 2nd-to-last number in the sequence has to be $2$. Now, substituting $2$ into the inverse formulae, if $I$ is even, $I=4$ (which is indeed even), and if $I$ is odd, $I=\frac{1}{3}$, which is not an integer. This means the 3rd-to-last number in the sequence has to be $4$. Substituting in $4$, if $I$ is even, $I=8$, but if $I$ is odd, $I=1$. Both of these are valid solutions, so the 4th-to-last number can be either $1$ or $8$. If it is $1$, then by the argument we have just made, the 5th-to-last number has to be $2$, the 6th-to-last number has to be $4$, and the 7th-to-last number, which is the first number, must be either $1$ or $8$. In this way, we have ultimately found two solutions: $N=1$ and $N=8$.

On the other hand, if the 4th-to-last number is $8$, substituting $8$ into the inverse formulae shows that the 5th-to-last number is either $16$ or $\frac{7}{3}$, but the latter is not an integer. Substituting $16$ shows that if $I$ is even, $I=32$, but if I is odd, $I=5$, and both of these are valid. If the 6th-to-last number is $32$, then the first number must be $64$, since $\frac{31}{3}$ is not an integer; if the 6th-to-last number is $5,$ then the first number has to be $10$, as $\frac{4}{3}$ is not an integer. This means that, in total, there are $4$ solutions for $N$, specifically, $1$, $8$, $10$, and $64$, which sum to $\boxed{\textbf{(E) }83}$.