REMINDERS: Answers in red.  Solvers (submitted/correct) in blue. (Forgive any omissions, but feel free to inform.) Comments in green. For further elaboration, please feel free to ask! 

1. Two is company and three is a crowd. 🙂   What are four and five?  9 (Nine).  Frank Green, Rita Barger, Barb Blue, Amy Ragsdale
2. a)  Find the smallest integer that is the product of 3 primes. 8 (2*2*2) Rita Barger, Barb Blue  b)  Same question for three different primes? 30 (2*3*5)  Frank Green, Rita Barger, Barb Blue, Amy Ragsdale
3. The sum of two numbers is 12 and their difference is 3.  What is the difference of the squares of the two numbers? 36  Frank Green, Barb Blue, Rita Barger, Amy Ragsdale
4. There are roughly 2 lbs of muscle for every 5 lbs of body weight. At that rate how much would the muscle of a 180 man weigh? 72 lbs. Frank Green, Rita Barger, Barb Blue, Amy Ragsdale
5. See the problem in the figure below.  Finding the answer seems frustrating and logic-defying at first (at least it was for me!), but when you see it, it’s like “DUH!  How’d I miss that?”  Indeed, non ‘math types’ probably get it quicker. 🙂  87 (We are viewing the numbers upside down.) Frank Green, Rita Barger, Barb Blue, Amy Ragsdale   

6. See the picture at right. –> In order to gain access here, what’s the minimum number of locks that need to be unlocked? [Two answers accepted here, depending on interpretation of what is seen (or not) in the pic.  🙂 ]  0 or 1.  Look at the lower horizontal bar.  It appears to some that neither of the two locks on that bar is actually affixed to the bar.  IF that’s the case, just remove either lock and pull out the horizontal bar, and then the vertical.  (No locks unlocked).  IF you think they are both affixed, then go to the upper horizontal bar, unlock ONE of the two locks there, and pull bar out opposite side (releasing the vertical bar). Don Hayes, Barb Blue, Rita Barger, Amy Ragsdale
7. Each side of a cube is increased by 10%. By what percent is the volume of the cube increased? 33.1 %.  Frank Green, Rita Barger, Barb Blue, Amy Ragsdale
8. A ‘message’ consists of a sequence of 3 dashes and 2 dots in some order. How many different (distinguishable) ‘messages’ are there? 10  Rita Barger, Barb Blue, Amy Ragsdale
9. When, if ever, is 1/x  > x?  [Reminder:  The symbol is ‘greater than’.] When x is between 0 and 1 OR less than -1.  Frank Green (Partial credit – Don Hayes, Rita Barger, Amy Ragsdale)
10. When, if ever, is | x | = – x? [Reminder:  Symbol is ‘absolute value’.] Whenever x is 0 or negative.  Don Hayes, Rita Barger, Amy Ragsdale.
11. What can be seen once in a minute, twice in a moment, and never in a thousand years? The letter ‘m’.  Frank Green, Rita Barger, Barb Blue, Amy Ragsdale
12. (#12 – #15 are re-shared.) It was bright and sunny at 12 noon today. What’s the probability it will be sunny again (in the same location) 60 hours from now? 0 (No chance).  60 hours from now, it will be midnight.  Frank Green, Rita Barger, Amy Ragsdale
13. Ralph drives 60 miles at an average speed of 30 mph, then returns over the same route at an average speed of 60 miles an hour. What was his average speed for the round trip? 40 mph. (120 miles traveled in 3 hours).  [You can’t average averages.]  Frank Green, Rita Barger, Amy Ragsdale
14. Find all the integers whose absolute value is less than 8, but more than 2. +/- 7, +/- 6, +/- 5, +/-4, and +/- 3 (10 integers in all.)  Rita Barger, Amy Ragsdale  (partial credit – Frank Green)
15. A tennis can is exactly the right size to hold three tennis balls stacked atop each other. Which is bigger, the can’s height (not counting lid) or its circumference (distance around)?  Why? The can’s circumference.  The can is exactly 3 ‘diameters’ high, but it is PI diameters around (formula for circumference).  Since PI is > 3, circumference is bigger.   Frank Green, Rita Barger, Amy Ragsdale
16. These last additions were added in the Mailing of Aug 24, 2020 (8/24/20).  How is that date related/connected to Feb 6, 2010? (Multiple answers possible?)  One answer:  8/24/20 could ‘reduce’ to 2/6/10. (Divide each number by 4).  No submissions. [For a GOOD reason!! I goofed!  – See BT#16 in Sep/Oct BTs.]  [The 2/6/10 should have been 2/6/05.  🙁  ]

BONUS 1:  See #3 above. The problem can be done directly without a lot of elbow grease.(trial/error, minor algebra, etc).   But, the problem is also somewhat of a ‘practical joke for math types’, as it has a fun, quick ‘mathy’ shortcut if you notice it. (You don’t even have to know what the numbers are!)  Can you tell what it is? A typical algebra student will first solve for the integers (4.5 and 7.5), then square each and subtract to get 36.  But if one notices the connection to the formula x^2 – y^2 = (x+y)(x-y), then one can just multiply 3*12 to get the 36.  🙂  Frank Green, Amy Ragsdale

BONUS 2:  The image at the right has been cropped from a larger cartoon about math majors as sporting fans.  If you’re a semi-serious math type, you may be slightly troubled.  Why? While it IS true that 0.9(repeating) actually = 1, that is NOT the case for 0^0.  In calculus, that’s an ‘indeterminate’ (undefined) form.  Don Hayes [Partial credit Amy Ragsdale]

BONUS 3:  In the July 13 Mailing, in the May/June BT Solutions (& Solvers),  I made a careless typo-like mistake listing an answer.  Can you find it?  (Hints:  Answers are in red font, the mistake is NON-mathematical, and the number of the problem is a multiple of 3.)  (As usual, a simple yes/no answer does not get full credit. 🙂 ).  In the solutions I had typed the Roman numeral for 19 as IXX, rather than XIX.  Amy Ragsdale  [Partial credit – Barb Blue]  Answer #2:  On Bonus #3, I typed ‘card’ rather than ‘car’ more than once. 🙁  Rita Barger.