NOTE: Newest BTs in red, Bonuses in blue, comments in green, updates in purple.
- Two is company and three is a crowd. 🙂 What are four and five?
- a) Find the smallest integer that is the product of 3 primes. b) Same question for three different primes?
- The sum of two numbers is 12 and their difference is 3. What is the difference of the squares of the two numbers?
- 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?
- See the problem in the figure below. Finding the answer seems frustrating and logic-defying at first (at least it was for me! Some are getting it immediately.), but when you see it, it’s like “DUH! How’d I miss that?” Indeed, non ‘math types’ probably get it quicker. 🙂 (Hint 1: You could drive into this very section of the parking lot in the ‘real world’ and not think a thing about it. Hint 2: Walk to the other side of the parking lot.)
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. 🙂 ]- Each side of a cube is increased by 10%. By what percent is the volume of the cube increased?
- A ‘message’ consists of a sequence of 3 dashes and 2 dots in some order. How many different (distinguishable) ‘messages’ are there?
- When, if ever, is 1/x > x? [Reminder: The symbol is ‘greater than’.]
- When, if ever, is | x | = – x? [Reminder: Symbol is ‘absolute value’.]
- What can be seen once in a minute, twice in a moment, and never in a thousand years?
- (#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?
- 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?
- Find all the integers whose absolute value is less than 8, but more than 2.
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? - 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?)
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?
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?
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. 🙂 ).
BT1…4 and 5 are 9. Or whole numbers, or positive integers, or some of the factors of 20 etc., or four-letter words.
BT3…The difference of their squares is 36.
BT4…A 180 pound man would have72 a muscle weight of 72lbs.
BT5…The car is parked in lot space 87. We see the space numbers upside down.
BT6…Only 2 locks are needed to be removed in order for the lid to be opened.
BT7…When each side of a cube is increased by 10%, the volume of the cube is increased by .331% if only computed to 3 decimal places. (However,if the original side length is 15, the newly increased volume is .33100148148…% higher,a
repeating decimal which can be rounded down to .331).
BT8…There are 17 “messages” that can be constructed from a collection of 3 dashes and 2 dots:CT,MD,GET,TIM,MEN,TAN,ANT,TP,MU,TUT,WA,WET,TX,EKE,GA,MR, and MEET.
Bonus#1…Absolute value of the difference of the squares of any two numbers is equal to the product of their sum and the absolute value of their difference
I am going to give this a shot again. As always, a lot of these are guesses.
1. What are 4 and 5? Nine? I really don’t know on this one. Four is a party and five is a mob?
2. The smallest integer that is the product of 3 primes? 8 (2 x 2 x 2)
3. The difference of the two squares is 36.
4. 2/5 of 180 is 72 pounds of muscle.
(I know, I should label my numbers/work.)
5. 87. I think I saw this one pretty fast. I remember thinking 16, 06, 68 . . ., but realized pretty quickly I should be reading from right to left.
6. No locks need to be unlocked/removed to gain access. Just pull out the bottom, horizontal rod and lift the vertical rod out from the top.
7. I always get increases and decreases by percentages wrong. I want to say the volume of the cube (when the sides are increased by 10%) is increased by 33.1%.
8. There are 9 (nine) distinguishable messages possible from the possible combinations of 3 dashes and 2 dots. Found that by trial and error. There has to be a better way.
Bonus 1: Sum is 12 and difference is 3. 12 x 3 = 36
Bonus 2: #0.999 . . . is essentially #1. I have no clue on the second one unless it involves trigonometry. The sine of 0 degrees is 1. But, no clue. So, yes, it troubles me, but the trouble is I don’t know the “why” it troubles me.
Bonus 3: Yes, I noticed the mistake. The Answer to question #8 is listed as #9.
As always, thanks for your work on finding these challenges. I will try to keep up with the new ones you add and I will keep thinking about these. I have been working on the ones that I missed (the added ones that I did not answer) from May/June before looking at the answers.
6. Zero:Slide the stainless steel pin on the bottom to the right, then slide the middle left-hand lock off the pin, and pull the pin to the right.
9. When x<1.
10. When x<0 or x=0.
B2. Calculus lists 0^0 as an indeterminate form.
BT2…30
BT9…when X is a negative integer other than -1.
BT10…never
BT11…the letter M
New Brainteasers:
9. When if ever is 1/x > x? Never unless negatives are somehow involved? But the question does not say -1 or -x, so never.
10. When is the absolute value of x equal to negative x? Never. The absolute value of a number is the distance from 0, but the distance does not have a designated direction. -5 is 5 “spaces” from 0 as is +5.
11. “M” is the answer. 1 in minute, 2 in moment, and none in a thousand years!
Isn’t math AMAZING?
BT12…There would be a zero % chance of it being sunny since 60 hours after 12 noon it would be 12 midnight.
BT13…A trip of 120 miles took 3 hours so the average speed was 40mph.
BT14…The integers would be 6,5,4,3,-3,-4,-6, and -6.
BT1…The circumference of the can would be a slightly larger number than the height because 2xPI is slightly more than the 6 used in determining the height..
BT12…There would be a zero % chance of it being sunny since 60 hours after 12 noon it would be 12 midnight.
BT13…A trip of 120 miles took 3 hours so the average speed was 40mph.
BT14…The integers would be 6,5,4,3,-3,-4,-6, and -6.
BT15…The circumference of the can would be a slightly larger number than the height because 2xPI is slightly more than the 6 used in determining the height..
1. 9 (4 + 5)
2. a) 8 (2x2x2) b) 30 (2x3x5)
3. 36
4. 72 lbs. of muscle
5. L8 (or an upside down 87)
6. minimum number of locks: one (according to my husband who is a mechanical genius!)
7. 33.1% increase in volume
8. 10 unique combinations (found by listing out possibilities so my confidence in this answer is not great!)
9. whenever 0 < x < 1
10. whenever x < 0
11. the letter m
12. 0% because it would be 12:00 midnight
13. 40 mph
14. -3, -4, -5, -6, -7 & 3, 4, 5, 6, 7
15. circumference is bigger because the height would = radius x 6, but the circumference would = radius x 2 x pi (which is slightly more than 6)
16. I'm stumped!
BONUS 1: The difference of the squares of the 2 numbers is always the same as the PRODUCT of the sum and difference of the numbers. (proved algebraically but it would be very difficult to type it all here – I saved the paper where I showed my work if you need to see it 🙂
BONUS 2: Neither guy's sign actually says "WE'RE #1, but the guy's sign on the right should say "WE'RE # cos 0 degrees".
BONUS 3: On #12 from the May/June BT answers, the Roman numeral for 19 was shown as IXX instead of XIX.