Jeremy has a box of crayons sitting on a shelf in a very dark room - too dark to distinguish crayon color. Inside this box, there are 21 red crayons, 13 blue crayons, and 6 brown crayons.
Now, Jeremy would like to go into the room and grab some crayons out of the box, without taking the whole box. But remember, the room is very dark. And the light doesn't work. And Jeremy's flashlight batteries are dead. And he has no replacement batteries. And he's not allowed to use matches....In other words, Jeremy will have to grab the crayons without seeing what colors they are.
Well, Jeremy is a very inquisitive boy, and this quest gets his restless mind working. "How many," he asks himself, "would I have to grab, if I want to make sure I get at least 2 crayons of the same color?" He thinks about this for a minute. "Obviously, I could just grab every crayon in the box, and then I'd know I have some of the same color." He shakes his head at the idea. "But I don't want to grab that many, because then they'll roll off the table, and I'll lose some... and of course little Jenny will eat some..."
Jeremy pauses at the door to the dark room. "Then again, I could just grab 2 crayons and hope that they're the same color. But what if they aren't? I really want 2 crayons of the same color. Oh, if only someone could tell me the smallest number that I absolutely have to grab in order to guarantee I have at least 2 crayons of the same color! That's the important thing - I need a GUARANTEE. Surely it doesn't have to be the whole box, does it?"
Question 1: What's the smallest number of crayons Jeremy needs to grab in order to guarantee that he has at least 2 crayons of the same color?
And if you get that, here are a few more!
Question 2: What if Jeremy wants to guarantee that he has at least 2 red crayons?
Question 3: What if Jeremy wants to guarantee that he has at least 1 crayon of each color?
Question 4: What if Jeremy wants to guarantee that he has more reds than browns?
Solution available here.
And if you still want more puzzles of a similar type, check out our "How Many Socks?" book, for sale at our store www.themathprofs.com. We'll walk you through the process of solving these types of questions - from questions just like these, slowly building to more general scenarios using variables for the numbers of colors and items.
Brian and Melanie Fulton both earned doctoral degrees in mathematics at Virginia Tech. They formerly taught math at the university level, and now run a hobby farm while accuracy-checking collegiate mathematics texts. They homeschool their four children, frequently employing the aid of chicken, dairy goat, cat, and dog tutors.