Sticker Sharing with Distinct Amounts
Ms. Rivera is preparing reward bags for five students: Ana, Ben, Chloe, Diego, and Ella. She has exactly 24 stickers to distribute. She wants each student to receive a different number of stickers, and every student must get at least 2 stickers. She also wants the student who receives the most stickers to get as many as possible, because that student has shown the most improvement. Before filling the bags, she pauses: what is the maximum number of stickers that the top student can receive? She also wonders whether the answer changes if she insists that the top student gets exactly 10 stickers. The stickers are identical, so only the counts matter. Ms. Rivera wants a clear justification, not just a guess, because she needs to explain her decision to the class. She has already written the names on five bags. The stickers come in a single sheet, so she can separate them freely.
1. What is the maximum number of stickers that one student can receive? Show how you find it.
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To maximize the top student's share, we must minimize the sum of the other four students' shares. Each must get at least 2 stickers and all five must get different numbers. The four smallest distinct numbers starting from 2 are 2, 3, 4, and 5. Their sum is 2+3+4+5=14. Subtract from 24: 24-14=10. So the top student can receive at most 10 stickers. A possible distribution is 2,3,4,5,10 (sum 24). Check that all are different and >=2. Yes. If we tried 11, the other four would need to sum to 13, but the minimum sum of four distinct numbers >=2 is 14, impossible. So 10 is maximum.
2. Explain why the other four students must receive the smallest possible distinct numbers to maximize the top share.
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Since the total is fixed at 24, making the top share larger means the remaining four shares must be as small as possible. The constraints are that each of the four gets at least 2 and all five numbers are distinct. To make the sum of the four as small as possible, we choose the four smallest distinct integers >=2: 2,3,4,5. Any other choice would replace one of these with a larger number, increasing the sum and leaving less for the top student. Therefore, the minimal sum for the four is 14, which gives the top student 10.
3. If Ms. Rivera insists that the top student gets exactly 10 stickers, list two different valid distributions of stickers among the five students.
Read the worked answer
With top=10, the other four must sum to 14, using four distinct numbers >=2. The only set of four distinct numbers >=2 that sum to 14 is {2,3,4,5}. So any distribution must use the numbers 2,3,4,5 for the other four students, and 10 for the top student. The two distributions differ only in which student gets which of the four smaller numbers. For example: Ana=2, Ben=3, Chloe=4, Diego=5, Ella=10; or Ana=5, Ben=4, Chloe=3, Diego=2, Ella=10. Both are valid and use each number once.