Jan/Feb17 Brain Teasers – SUMMARY
CALENDAR WINNERS: Six different submitters earned a total of 106 entries by submitting answers to the Brain Teasers. Winners of the drawing were Jennifer Steele (Desk or Wall Calendar) and Jim Waterman (Desk Calendar)
SOLUTIONS: See below.
REMINDER: Answers in red. Solvers (submitted/correct) in blue. (Forgive any omissions.) Comments in green. For further elaboration, feel free to ask!
1A. There are several interesting facts/traits surrounding the number 2017. Share one or more that you find. (NO ‘wrong’ answers here!) Lots of correct answers here. Most noted that 2017 is prime. Connie Dewey, Amy Ragsdale, Anita Dixon, Jennifer Steele Anita Dixon noted (in the spirit of Ramanajuan) that 2017 can be written as the sum of 5 different cubes in (at least) 2 ways, and gave both ways!
1B. One special date in 2017 is 1/17/17. It is the only “Year-Product-Day” [YPD] this whole year! Can you define (or guess) what a YPD is? A date where the month*day = year (two digits) Connie Dewey, Anita Dixon. If so, when will the next one occur? 1/18/18. Connie Dewey, Anita Dixon When is the next year with only one YPD? 2019. Anita Dixon When is the first year this century without a YPD? (Hint: We haven’t had it yet, if at all.) 2037. Connie Dewey, Anita Dixon
2. Find a 5-digit number where the sum of all the digits is 25, and each digit is 2 more than the digit preceding it. 13579; Dan Felshin, Connie Dewey, Amy Ragsdale, Anita Dixon, Jim Waterman.
3. What number is 20 more than twice the number that is 10 more than 15 times one-half of ten? 190; Dan Felshin, Connie Dewey, Amy Ragsdale, Anita Dixon, Jennifer Steele.
4. Counterfeit Coin, Part 1 You have 12 coins in front of you, knowing that one of them is counterfeit. All you know is that the ‘bad’ coin is slightly heavier than the others. You also have a balance scale in front of you, and your task is to determine the ‘bad’ coin using only three (3) weighings*. How do you accomplish this? Weighing 1: Split the coins into two piles of 6, and balance them against each other. The heavier group contains the bad coin, so use it next. Weighing 2: Split this group into two piles of 3 and balance them against each other. Weighing 3: Take any two of the heavy group and put them on each side of the scale. a) If they balance, the unused coin is the counterfeit. b) If they don’t balance, the heavier side is the counterfeit. Dan Felshin, Amy Ragsdale, Connie Dewey, Jennifer Steele, Jim Waterman.
5. Miss Demeanor often goes the wrong way on a one-way street – RIGHT PAST a police station. How does she manage to do this without ever getting a ticket? She was walking! (or riding a bike, or skateboarding . . . 🙂 ). Jim Waterman, in his usual way, included FIVE (5) other, less-likely-but-perfectly-acceptable answers. (To see them, go to Jan/FebBTs and check the (now posted) comments.) Amy Ragsdale, Anita Dixon, Jennifer Steele, Jim Waterman
6. Find three consecutive integers that have the same product as sum. There are (at least?) three possible triplets that work: (1,2,3), (-1,0, 1), and (-3, -2, -1). Connie Dewey, Amy Ragsdale, Anita Dixon, Jim Waterman
7. Find a fraction which is equivalent to 1/2 and whose numerator and denominator have a product of 72. 6/12. Connie Dewey, Amy Ragsdale, Anita Dixon, Jennifer Steele, Jim Waterman
8. Find the smallest positive integer which is divisible by each integer from 1 to 10 inclusive. 2520 (=2*2*2*2*3*5*7) Connie Dewey, Amy Ragsdale, Anita Dixon
9. Fill in the missing letters: I V X __ ___ D M. L & C. (Roman numerals). AND, my initials! 🙂 Amy Ragsdale, Anita Dixon, Jennifer Steele, Jim Waterman
10. Consider a 7 x 7 x 7 stack of smaller cubes, all of the same size. How many of these are totally hidden from view? 125 (5 x 5 x 5). Anita Dixon, Jim Waterman
11. You are facing south. You turn left, make an about-face, turn right, and then turn right again. Which direction is behind you? West. Amy Ragsdale, Anita Dixon, Jennifer Steele
12. (All names are street names.) Belvedere is parallel to St. Anthony. Davidson is perpendicular to River, which is parallel to St. Anthony. Is Davidson parallel or perpendicular to Belvedere? Perpendicular. Amy Ragsdale, Anita Dixon, Jennifer Steele, Jim Waterman
BONUSES
NOTE: Four (4) subscribers have earned a total of 13 entries for the Rollin’ Down The River book later this Spring. There entries are tabulated and will be carried forward.
A. Counterfeit Coin – Part 2 Same situation as #4 above. Twelve coins, one is counterfeit, but you don’t know if it’s too light OR too heavy. You still have (only) 3 weighings to determine the bad coin. NOTE: It can be done, but it takes some clever thinking. (Solutions will need to explain carefully for full credit.) [3] There were no correct (or even submitted) answers here. To see a solution, go to Jan/FebBTs , and check out the latest comment from Dan Felshin for a fun You-Tube solution (thanks, Dan).
B. (See #1B above for details.) What is the only year (last two digits) each century that has seven (7) Year-Product Days? [2] Anita Dixon submitted the correct answer here (and has received 2 entries), but this Bonus will be carried forward one to the Mar/Apr BTs.
C. (See #8 above). Find the corresponding smallest integer divisible by each integer from 1 – 20. [2] 232,792,560. (=2*2*2*2*3*3*5*7*11*13*17*19) Amy Ragsdale, Anita Dixon. (Originally re-posted answer to#8 here. Thanks to Anita Dixon for catching error.)
D. In #10 above, how many of the cubes have only 1 face visible? [2] 150 (25 on each of 6 faces). Anita Dixon. (Jennifer Steele also received one entry for a submitted solution.)
E. Here’s a fun BT I posted a year or so ago, and didn’t get too many nibbles. It’s not as hard as it might look, so I’ll pitch it out again. See 3LogiciansBar. [2] For EACH of the first two gals: Since neither could know about the gal(s) on their left, they had to answer “I don’t know” to the question of whether they all 3 wanted a beer. The last gal, deducing this (and knowing that if either of the first two had NOT wanted a beer, they would have answered “No”), was able to answer “Yes” (since she wanted a beer too – or SHE would have said NO). Anita Dixon, Jim Waterman
