Figure This! provides interesting mathematical challenges for that middle-school students can do at home with their families.
Each challenge features:
- a description of the important math involved
- a note on where the math is used in the real world
- a hint to get started
- complete solutions
- a "Try This" section
- additional related problems with answers
- questions to think about
- fun facts related to the math
- resources for further exploration.
Here's a sample challenge: #28 Pigeonholes
Did You Know That? The pigeonhole principle was so named because if 10 homing pigeons return to 9 holes, then at least one hole must have two pigeons in it.
Figure this! How many people would have attend a school like Seminole Ridge High (student count of 2448) before it contained at least two students with the same first and last initials?
Hint: Consider a simpler problem. How many people would have to enter a classroom before it contained at least two students with the same first initial?
Get Started: How many different possibilities are there for a first initial? for a last initial? for both initials combined?
Solution is in the comments.
There are 26 letters in the English alphabet. So there are 26 different possibilities for the first initial. Consider all the possible pairs of two initials.
ReplyDeleteFor example, suppose a person has the first initial A. Then the pair of initials could be AA, AB, AC, ., AZ. There are 26 different possibilities. If the first initial is B, the pair of initials could be BA, BB, BC, ., BZ. Again there are 26 different pairs.
Continuing in this way and since there are 26 possible first initials, each of which could be paired with 26 last initials, there are 26 × 26, or 676 possible different pairs of initials. If there were 677 people, at least two of them must have the same pair of initials.