Daily Math Puzzle: 2026-07-01
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2026-07-01
At 'The Confectionery Corner', artisanal Dark Delights cost $X each and Milk Marvels cost $Y each, where X and Y are distinct positive whole dollar amounts. A customer bought the exact same number of Dark Delights as Milk Marvels, and the total bill came to exactly $27. Which of the following statements about the number of chocolates of each type purchased *must be true*?
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Solution
The number of chocolates of each type must be an odd number. — Let 'N' be the number of Dark Delights purchased (and also Milk Marvels purchased). Let X be the price of Dark Delights and Y be the price of Milk Marvels.
The total bill is N * X + N * Y, which can be factored as N * (X + Y). We are given that the total bill is $27.
So, N * (X + Y) = 27.
We are also told that X and Y are distinct positive whole dollar amounts. The smallest possible value for X is $1 and the smallest possible value for Y (distinct from X) is $2. Therefore, the sum (X + Y) must be at least 1 + 2 = 3.
Now, let's consider the factors of 27: 1, 3, 9, 27.
Since N * (X + Y) = 27, N must be one of these factors.
However, we know (X + Y) >= 3. This means that if N = 27, then (X + Y) would have to be 1 (because 27 * 1 = 27), which contradicts our condition that (X + Y) must be at least 3.
So, N cannot be 27.
The possible values for N are therefore 1, 3, or 9.
Now, let's evaluate the given choices based on these possible values for N:
1. "The number of chocolates of each type must be a prime number."
- N=1 is not a prime number.
- N=9 is not a prime number.
Therefore, this statement is FALSE.
2. "The number of chocolates of each type must be an odd number."
- N=1 is an odd number.
- N=3 is an odd number.
- N=9 is an odd number.
Therefore, this statement is TRUE for all possible values of N.
3. "The number of chocolates of each type must be a perfect square."
- N=3 is not a perfect square.
Therefore, this statement is FALSE.
4. "The number of chocolates of each type must be greater than 3."
- N=1 is not greater than 3.
- N=3 is not greater than 3.
Therefore, this statement is FALSE.
Thus, the only statement that *must be true* is that the number of chocolates of each type is an odd number.
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