Snake Weight System of Equations

Worked Solution

How to Think About It: Three parts — head, body, tail. The head is pinned directly (12 oz); the body and tail are each defined *relative* to the others, so this is a small linear system. The heuristic is substitution: express the two unknowns in terms of one, collapse to a single equation, solve. The trap is trying to guess-and-add without noticing that the body is defined against the *head+tail combined*, not against the whole snake — read the dependency chain before writing anything.

Quick Estimate: Do the fixed-point in your head in two moves. The body is $1.5\times(\text{head}+\text{tail})$, and the tail is half the body — so the tail feeds back into the body's own definition. Ignore the tail for a first pass: body $\approx 1.5\times 12 = 18$. Now the tail is about half of that, $\approx 9$, which bumps the head+tail combined to $\approx 21$, so body $\approx 1.5\times 21 \approx 31$; tail climbs to $\approx 16$, combined $\approx 28$, body $\approx 1.5\times 28 \approx 42$... the iteration is still climbing, which tells you the naive $18$ was a big underestimate. Close it exactly: the tail is $0.75$ of the *body* worth of feedback, so $b = 18 + 0.75b \Rightarrow 0.25b = 18 \Rightarrow b = 72$. Then tail $= 36$, total $= 12 + 72 + 36 = 120$ oz. The lesson of the estimate: the self-referential tail term roughly *quadruples* the body from the $18$ first guess.

Approach: Set up two equations, substitute the tail into the body.

Formal Solution:

Let $b$ = body weight, $t$ = tail weight (ounces); head $= 12$.

$$t = \tfrac{1}{2}b, \qquad b = 1.5\,(12 + t).$$

Substitute the first into the second:

$$b = 1.5\left(12 + \tfrac{b}{2}\right) = 18 + 0.75\,b \;\Rightarrow\; 0.25\,b = 18 \;\Rightarrow\; \boxed{b = 72}.$$

Then $t = 72/2 = 36$, and

$$\text{total} = 12 + 72 + 36 = \boxed{120 \text{ oz}}.$$

Answer: The snake weighs 120 ounces.