1. **Problem statement:** We have three players: Archie, Evelyn, and Jacob. Initially, their counters are in the ratio 7 : 5 : 6. At the end, the ratio changes to 4 : 3 : 5. We know Archie ends with 12 counters. We need to find how many counters Archie started with. 2. **Understanding ratios and total counters:** Let the initial counters be $7x$, $5x$, and $6x$ for Archie, Evelyn, and Jacob respectively. Let the final counters be $4y$, $3y$, and $5y$ respectively. 3. **Total counters remain constant:** The total counters at the start and end are equal: $$7x + 5x + 6x = 4y + 3y + 5y$$ $$18x = 12y$$ 4. **Express $y$ in terms of $x$:** $$y = \frac{18x}{12} = \frac{3x}{2}$$ 5. **Use Archie's final counters to find $x$:** Archie ends with 12 counters, so: $$4y = 12$$ Substitute $y$: $$4 \times \frac{3x}{2} = 12$$ $$6x = 12$$ $$x = 2$$ 6. **Find Archie's initial counters:** $$7x = 7 \times 2 = 14$$ **Answer:** Archie started the game with 14 counters.