Earlier today, I provided you with five unique puzzles from Mathigon’s advent calendar. In case you missed them, here they are again, along with their solutions. My apologies if you were looking for information on Ian Dury. However, you now have the opportunity to engage your mind in these challenges:
1. "Hit me with your four cube stick"
Solution:
2. "Timelines"
Solution: The hands on a clock face align and form a straight line twenty-two times every 24 hours. Assuming 6:00 a.m. marks the first time, the hands will align every one hour, five minutes, and twenty-seven seconds.
3. "Eight heavy adult females"
Solution: If you add up the numbers 888, 88, 8, 8, and 8, it equals 1,000. To make 100, you can add together 88 + 8 + ((8 + 8 + 8 + 8)/8), or you can use this formula: 8 + ((8 + 8)/8) + 8 + 8 – 8 – 8.
4. "Slice of Orange"
Solution: The answer for this puzzle is π/6. If C represents the center of the semicircle and r represents the radius, you can create a right-angled triangle, shaded yellow, that has two sides of length r. Using Pythagoras’ theorem, the hypotenuse of this triangle is r√2. We can also use Pythagoras’ theorem for the red triangle, OCE, which has a hypotenuse of 1. If (r√2)^2 + r^2 = 1/2, then 3r^2 = 1, so r = 1/√3. The area of a circle of radius r is πr^2, so the area of a semicircle of radius r is πr^2/2. The answer is, therefore, π(1/√3)^2/2 = π/(3 x 2) = π/6.
5. "Bucket List"
Solution: The answer to this puzzle is 1200/3125, or 38.4%. There are 5^5 = 3125 total possible arrangements (five balls in five buckets). To have a single empty bucket, only one other bucket must have 2 balls while each of the remaining three has 1 ball. There are 5 ways to pick an empty bucket and 4 ways to pick the bucket with two balls. There are "5-choose-2" = 10 ways to pick the two balls that get paired together in the same bucket, and 3 x 2 = 6 ways to distribute the remaining three balls in the remaining three buckets. Thus, the total number of arrangements with one empty bucket is 5 x 4 x 10 x 6 = 1200. The probability is 1200/3125, which equals 38.4%.
Thank you to Mathigon for presenting today’s puzzles. Mathigon is a remarkable, free website about math with excellent interactive tools. I highly recommend it!
I post a new puzzle every other Monday. If you wish to suggest an exciting puzzle, please contact me via email. I also hold school talks discussing math and puzzles, both online and offline. If your school is interested, please contact me.
