In this article, we explore the problem of determining the least number of coins required to make any change from 1 to 99 cents. This is a classic problem in the field of combinatorics and optimization, often encountered as a subproblem in dynamic programming.

Understanding the Coin Change Problem

The goal is to find a minimal set of coin denominations, which can efficiently fulfill any request for change between 1 and 99 cents. In standard US currency, the available denominations are 1-cent (penny), 5-cents (nickel), 10-cents (dime), 25-cents (quarter), and 50-cents (half-dollar).

To solve this problem, we need to ensure that for any amount of change from 1 to 99 cents, we can cover it using the fewest possible number of these coins.

Greedy Algorithm Approach

A greedy algorithm is an instinctive approach to solve the problem, where we try to use the largest denomination coins first. This approach is optimal for certain sets of denominations like the typical US coins but does not work universally for all sets of denominations. Here's how it applies to US coins:

1. Start with the largest denomination.
2. Use as many of this denomination as possible without exceeding the change amount.
3. Move to the next largest denomination. Repeat until the exact change is obtained.

Example

Suppose you need to make 41 cents. Using the greedy algorithm approach:

1. Use one 25-cent coin.
2. Then use one 10-cent coin to make a total of 35 cents.
3. Next, use a 5-cent coin, totaling 40 cents.
4. Finally, use a 1-cent coin to make up to 41 cents.

This method uses a total of 4 coins.

Calculating Minimum Coins for All Amounts from 1 to 99 Cents

To ensure that any change can be made efficiently, let's calculate the minimum number of coins required for each amount from 1 to 99 cents. The table below summarizes the key points:

Dynamic Programming Solution

A dynamic programming approach provides a general and efficient way to solve the problem, particularly for non-standard sets of denominations. The essence is to build a `dp` array where `dp[i]` represents the minimum number of coins needed to make `i` cents.

Dynamic Programming Algorithm:

1. Initialize an array `dp` with size 100, with `dp[0] = 0` and all other values set to infinity.
2. For each denomination `coin`, update the dp array: • For each amount `i` from the denomination `coin` to 99, calculate `dp[i] = min(dp[i], dp[i - coin] + 1)`.
3. The value of `dp[99]` will give the minimum number of coins needed for 99 cents.

Mathematical Formulation

The formulation of the dynamic programming solution can be expressed as:

$$dp[i] = \begin{cases} 0, & \text{if } i = 0 \\ \min{dp[i], dp[i - \text{{coin}}] + 1}, & \text{for each coin and } i \geq \text{{coin}} \end{cases}$$

Conclusion

After evaluating all possible combinations using the US standard denominations, the least number of coins needed to make any change from 1 to 99 cents is 9 coins. You can verify this with simulations or computational verifications using either a greedy approach or dynamic programming. The greedy algorithm, thanks to its efficiency for US denominations, will suffice to find this solution for real-world applications.

By employing such strategies, solutions to similar coin change problems can help in efficient cash handling for businesses, automated machines, and economic studies. This minimalistic approach also demonstrates early instances of optimization in computational theory.