What is the Least Amount of Coins the Amount Needs to Be the Same for All Combinations That Have 7 Different Combinations and Also Adds Up to a Dollar?

This question explores the intricacies of using different combinations of quarters, dimes, nickels, and pennies to sum to exactly one dollar (1.00) while minimizing the total number of coins used in exactly seven distinct combinations. This problem can be solved using a brute force approach with the J programming language.

Brute Force Solution Using the J Programming Language

To determine the smallest number of coins that can form exactly seven distinct combinations totaling one dollar, we used the J programming language (similar to APL) to find all possible combinations of quarters, dimes, nickels, and pennies. This brute force method systematically identifies all unique combinations of these coins that sum to the exact value of one dollar.

Step-by-Step Brute Force Approach

Identify All Possible Combinations: Use a loop or recursive method to generate all possible combinations of quarters (25 cents), dimes (10 cents), nickels (5 cents), and pennies (1 cent). Filter Valid Combinations: From the generated combinations, filter to keep only those that add up to exactly one dollar (100 cents). Count the Valid Combinations: After filtering, count the number of unique combinations that sum to one dollar. In this case, there are 242 such combinations. Find the Coins in Each Combination: For each combination, count the number of coins used. Histogram of Coin Counts: Create a histogram to find the number of times each coin count appears exactly seven times in the 242 valid combinations. Identify Coin Sets with Exact Seven Combinations: Use the histogram to identify the coin sets that have exactly seven members, each of which sums to one dollar.

Results

The results show that the minimum number of coins required to form exactly seven distinct combinations totaling one dollar is 16 coins. We identified four groups of coin sets that meet this criterion, each containing exactly seven members:

16 coins 19 coins 24 coins 28 coins

Among these, the group with 16 coins is the smallest, and each member in this group sums to exactly one dollar.

The 16-Coin Combinations

The 7 sets of 16 coins that sum to one dollar are:

0 quarters, 4 dimes, 12 nickels, 0 pennies 0 quarters, 8 dimes, 3 nickels, 5 pennies 1 quarter, 0 dimes, 15 nickels, 0 pennies 1 quarter, 4 dimes, 6 nickels, 5 pennies 2 quarters, 0 dimes, 9 nickels, 5 pennies 2 quarters, 4 dimes, 0 nickels, 10 pennies 3 quarters, 0 dimes, 3 nickels, 10 pennies

Pro Gamer Scruffy's Solution

Pro Gamer Scruffy provided another insightful but simplified approach to solving this problem. According to Scruffy, to find the smallest amount of coins with exactly seven combinations totaling one dollar, start by using the largest coin, which is a quarter (25 cents).

With one quarter, only one combination is possible (4 quarters). To increase the number of combinations, add other coins:

Add one dime (10 cents) to get 6 more combinations (1 to 6 dimes in combination with the quarter). Add nickels (5 cents) and pennies (1 cent) to further increase the combinations.

In the end, the optimal combination that provides exactly seven distinct combinations is 11 coins – 3 quarters, 2 dimes, 1 nickel, and 5 pennies. These 11 coins can form exactly seven different combinations, all summing to one dollar.

Conclusion

In summary, while both the brute force method with the J programming language and Pro Gamer Scruffy's simplified method provide valid answers, the brute force approach offers a detailed and systematic breakdown of the problem. Pro Gamer Scruffy's method, on the other hand, provides a more intuitive understanding of why specific coin combinations work.