Lisa has 10 friends and 34 marbles. What is the minimum number of additional marbles she needs so that she can give each friend at least one marble and no two friends receive the same number of marbles?
Lisa wants to minimize the number of marbles she gives to her friends without giving any two of them the same number of marbles. The minimum number of marbles she can give to a friend is 1. She then gives 2 marbles to another friend, then 3 to another, then 4, and so on, until the last friend receives 10. The total number of marbles Lisa has given away is $1+2+3+\cdots+10 = \frac{10 \cdot 11}{2}=55$.

Thus, Lisa needs $55-34=\boxed{21}$ more marbles.
21
Level 3
Algebra
Problem: Lisa has 10 friends and 34 marbles. What is the minimum number of additional marbles she needs so that she can give each friend at least one marble and no two friends receive the same number of marbles?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine the minimum number of additional marbles Lisa needs, we first recognize that she has 10 friends. Each friend must receive a unique number of marbles, starting from at least 1 marble for the first friend. The smallest unique distribution of marbles for 10 friends is 1, 2, 3, ..., up to 10. 

The sum of the first n natural numbers can be calculated using the formula: 

Sum = n(n + 1) / 2.

For n = 10, the sum becomes:

Sum = 10(10 + 1) / 2 = 10 * 11 / 2 = 55.

This means Lisa needs a total of 55 marbles to give each of her 10 friends a unique number of marbles. 

Currently, Lisa has 34 marbles. To find out how many additional marbles she needs, we subtract the number of marbles she currently has from the total required:

Additional marbles needed = Total required - Current marbles = 55 - 34 = 21.

Thus, the minimum number of additional marbles Lisa needs is 21.<|END_OF_LATENT|> The final answer is $21$.