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Mia has a collection of 50 unique stickers that she wants to share with her friends. She decides to put the stickers into envelopes to give out, following these rules:
1. Each envelope must contain at least one sticker.
2. No two envelopes can contain the same number of stickers.
3. All stickers must be distributed.
What is the maximum number of envelopes Mia can use to distribute all her stickers?
Let’s solve this problem step-by-step:
1. To maximize the number of envelopes, we should start with the smallest possible numbers of stickers per envelope.
2. Let’s start distributing:
Envelope 1: 1 sticker
Envelope 2: 2 stickers
Envelope 3: 3 stickers
And so on…
3. Let’s continue this pattern until we can’t add another envelope without exceeding 50 stickers:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 = 45 stickers (9 envelopes)
4. We’ve used 45 stickers in 9 envelopes. We have 5 stickers left to distribute.
5. We can’t create a 10th envelope with 5 stickers because we already have an envelope with 5 stickers.
6. The optimal solution is to add these 5 stickers to existing envelopes, one each to the last 5 envelopes:
Envelope 1: 1 sticker
Envelope 2: 2 stickers
Envelope 3: 3 stickers
Envelope 4: 4 stickers
Envelope 5: 6 stickers (+1)
Envelope 6: 7 stickers (+1)
Envelope 7: 8 stickers (+1)
Envelope 8: 9 stickers (+1)
Envelope 9: 10 stickers (+1)
Therefore, the maximum number of envelopes Mia can use is 9.