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Bunuel
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I32-07 12 Mar 2026, 07:00
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Dropdown 1: 5 Dropdown 2: 1 empty lipstick


A drugstore chain can return empty lipsticks that it originally sold to the supplier for recycling. To encourage customers to return their empty lipsticks to its stores rather than throw them in the trash, the drugstore chain has instituted a new policy allowing customers to bring empty lipsticks to a store and exchange them for a new, full lipstick, as shown in the diagram.

Select from each drop-down menu the option that creates the most accurate statement based on the information provided.


Dara has 11 empty lipsticks originally purchased from this drugstore chain. If Dara exchanges these for new lipsticks under the store policy, wears the new lipsticks until they run out, and then repeats these actions (with the lipsticks that are now empty) to the fullest extent possible—without acquiring any additional lipsticks outside of the exchange program—then Dara will be able to acquire and wear new lipsticks, and will have upon reaching “End”.
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Dropdown 1: 5 Dropdown 2: 1 empty lipstick
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Bunuel
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Official Solution:
Bunuel



A drugstore chain can return empty lipsticks that it originally sold to the supplier for recycling. To encourage customers to return their empty lipsticks to its stores rather than throw them in the trash, the drugstore chain has instituted a new policy allowing customers to bring empty lipsticks to a store and exchange them for a new, full lipstick, as shown in the diagram.

Select from each drop-down menu the option that creates the most accurate statement based on the information provided.


Dara has 11 empty lipsticks originally purchased from this drugstore chain. If Dara exchanges these for new lipsticks under the store policy, wears the new lipsticks until they run out, and then repeats these actions (with the lipsticks that are now empty) to the fullest extent possible—without acquiring any additional lipsticks outside of the exchange program—then Dara will be able to acquire and wear new lipsticks, and will have upon reaching “End”.
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The store’s program allows Dara to trade 3 empty lipsticks for 1 new full one, as shown by the “Yes” path at the first question. With 11 empty lipsticks to begin with, Dara will be able to make this exchange three times (following the “Yes” arrow and then looping back, three times each). In so doing, Dara will trade 9 empty lipsticks for 3 full ones—and thus will have 3 full and 2 empty lipsticks.

At this point, Dara will not be able to make another exchange. So Dara wears one of the new lipsticks until it runs out, at which point Dara will have 2 full and 3 empty lipsticks, and will have worn 1.

Dara can now trade those 3 empty lipsticks for an additional full one—ending up with 3 full lipsticks (and no empties).

At this point, Dara will wear all three of these lipsticks one by one. After wearing all three, Dara will have just the 3 empty lipsticks, and will now have worn 4 lipsticks total.

Dara can then trade those 3 empties for one last full lipstick, and then wear that lipstick—the fifth and final one that Dara can wear. Dara will then reach “End” with the empty container of this one last lipstick. So, Dara acquires and wears 5 lipsticks total, and reaches “End” wih 1 empty lipstick.


Correct answer:

Dropdown 1: "5"

Dropdown 2: "1 empty lipstick"
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I32-07 19 Apr 2026, 05:55
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Hi Bunuel,

can we use following shortcut to answer this question

When:
  • k empty items → 1 full item
  • And every full item eventually becomes empty
  • And nothing is added from outside
👉 Net effect of 1 full‐use cycle
Each new lipstick consumes (k − 1) empties permanently.
Why?
  • To get 1 full lipstick: you give k empties
  • After using it, you get 1 empty back
  • Net loss = k − 1 empties

Apply to This Problem
Given
  • Exchange rate = 3 empty → 1 full
  • Net loss per new lipstick = 3 − 1 = 2 empties
  • Starting empties = 11

Step 1: Maximum new lipsticks possible

Max new lipsticks=⌊Starting empties/(k−1)⌋=⌊11/2⌋=5
✅ Dara can acquire and wear 5 new lipsticks

Step 2: Final empties left
Final empties=11−(5×2)=1
✅ 1 empty lipstick remains

✅ Final Answer (using shortcut)
  • New lipsticks acquired and worn: 5
  • Empty lipsticks at End: 1

Bunuel
Official Solution:


The store’s program allows Dara to trade 3 empty lipsticks for 1 new full one, as shown by the “Yes” path at the first question. With 11 empty lipsticks to begin with, Dara will be able to make this exchange three times (following the “Yes” arrow and then looping back, three times each). In so doing, Dara will trade 9 empty lipsticks for 3 full ones—and thus will have 3 full and 2 empty lipsticks.

At this point, Dara will not be able to make another exchange. So Dara wears one of the new lipsticks until it runs out, at which point Dara will have 2 full and 3 empty lipsticks, and will have worn 1.

Dara can now trade those 3 empty lipsticks for an additional full one—ending up with 3 full lipsticks (and no empties).

At this point, Dara will wear all three of these lipsticks one by one. After wearing all three, Dara will have just the 3 empty lipsticks, and will now have worn 4 lipsticks total.

Dara can then trade those 3 empties for one last full lipstick, and then wear that lipstick—the fifth and final one that Dara can wear. Dara will then reach “End” with the empty container of this one last lipstick. So, Dara acquires and wears 5 lipsticks total, and reaches “End” wih 1 empty lipstick.


Correct answer:

Dropdown 1: "5"

Dropdown 2: "1 empty lipstick"
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