AMC 8 2020 Problem 22 | Number Game Problem

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

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.

$7 \rightarrow 22 \rightarrow 11 \rightarrow 34 \rightarrow 17 \rightarrow 52\\ \rightarrow 26$

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 $N$ ?

$N \rightarrow \longrightarrow \rightarrow-\rightarrow \longrightarrow \rightarrow 1$

• 73
• 74
• 75
• 82
• 83

AMC 8 2020 Problem 22

83

Try to start with the final output and work backwards.

Try to form a tree keeping in mind all the possible outcomes.

So, the sum will be,

$1+8+64+10=83$

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