# Cookie Box Mix-up

Mrs. V wants to give cookies to the 7 students in her class, so she orders some from Bob's Cookie Emporium. Due to a traumatic childhood incident, she strongly desired the delivery to be divisible by 7 so she can fairly split them among her students (and due to further childhood trauma refuses to eat some herself, trash them, or otherwise reduce the number of delivered cookies to make things work out.)

Bob assures her that every box has a consistent number of cookies, and that four small boxes and one large box will yield a number of cookies divisible by 7. Satisfied, Mrs. V orders.

However, when they arrive, the order was mixed up and instead she received one small box and two large boxes. She started to scream and the cookie delivery person, but before her obscenities became too vulgar, the student Samantha calmly said "Don't worry Mrs. V! The number of cookies must still be divisible by 7." How did Samantha know?

• I'm not sure this is a puzzle. The math is pretty straightforward.
– xnor
Apr 20, 2015 at 3:58
• Just because it's not a hard puzzle hardly makes it not a puzzle =P Apr 20, 2015 at 4:05

One of the correct order and three of the incorrect contain, in total, 7 small boxes and 7 large box, and so be divisible by 7. Since the correct order is divisible by 7, then so is 3 of the incorrect order. Therefore, the incorrect order is also divisible by 7.

For a more straightforward algebraic solution, working modulo 7,

$$S+2L = 8S+2L = 2(4S+L) = 0$$

since $4S+L$ is a multiple of $7$.