1. **Problem Statement:** We have three types of chocolate bags containing 6, 9, and 20 chocolates respectively. We want to find the largest number of chocolates that cannot be obtained by buying any combination of these bags. 2. **Key Concept:** This is a classic problem related to the Frobenius coin problem or the Chicken McNuggets theorem, which deals with finding the largest number that cannot be expressed as a non-negative integer combination of given numbers. 3. **Step-by-step Explanation:** 1. We want to find the largest integer $N$ such that $N \neq 6a + 9b + 20c$ for any non-negative integers $a,b,c$. 2. Since 6 and 9 share a common divisor 3, any combination of 6 and 9 alone can only produce multiples of 3. 3. The number 20 is not divisible by 3, so including 20 allows us to reach numbers not divisible by 3. 4. The problem shows that all numbers greater than 43 can be formed by some combination of 6, 9, and 20. 5. To prove 43 cannot be formed, we try to express 43 as $6a + 9b + 20c$: - Since 43 mod 3 is 1, and 6 and 9 combinations are multiples of 3, at least one 20 must be used. - Subtract 20 once: $43 - 20 = 23$ - Subtract 20 again: $23 - 20 = 3$ - Now, 3 cannot be formed by 6 or 9 (both multiples of 3 but minimum is 6), so 43 is unreachable. 6. Therefore, the largest unreachable number is 43. 4. **How to solve similar problems in exams:** - Identify the numbers you can combine. - Check if they have a common divisor; if yes, numbers not divisible by that divisor might be unreachable. - Try to express numbers starting from a certain point to see if all larger numbers can be formed. - Use the idea of modulo classes to group numbers and check reachability. - Test the largest unreachable number by attempting to express it as a combination. **Final answer:** $$\boxed{43}$$