SUMMARY: Solutions & Solvers, Mar/Apr BTs

by | Apr 24, 2022 | Various Answers | 2 comments

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

  1. If all of the integers from 1 – 1000, inclusive were written out – and then arranged in alphabetical order – what would be the very last entry in the list? (Hint:  This one’s in honor of a recent specially-celebrated numerical day.  Thanks to Millie Johnson for the idea.)  Two (hundred) twenty-two.  (In honor of 2/22/22.) Frank Green, Rita Barger, Alexis Avis, Jim Waterman, Anita Dixon
  2. One half of N plus one half of N equals 5. Find the value of N. N = 5. Frank Green, Rita Barger, Alexis Avis, Anita Dixon
  3. Consider this string: 8 2 7 5 6 4 5 3 4.  If the difference between the first number and the fifth number is greater than the difference between the first number and the sixth number, respond with the number 7.  Otherwise, take the difference between the differences and respond with that number.  2. Frank Green, Rita Barger, Alexis Avis, Anita Dixon
  4. In Boontown, streets that begin with a vowel run east-west, unless they also end with a vowel, in which case they run north-south.  Other streets can go either way.  Berkeley street is perpendicular to Alice street.  In which direction does Berkeley run? East-West. Frank Green, Rita Barger, Jim Waterman, Anita Dixon
  5. A customer in a restaurant found a dead fly in his coffee.  He sent the waiter back for a fresh cup.  After a sip he shouted “This is the same cup of coffee I had before!”  How did he know? One possible reason:  He had sugared his coffee and it was still sugared.  Frank Green, Rita Barger, Jim Waterman
  6. Have we done this one before?  Can’t remember, but, either way,  have fun!.  (NOTE:  There are actually two different answers here [see Bonus #4], but for this BT, let’s assume that the speaker is female, as the BT seems to do.) C. Daughter.  Anita Dixon.                                                                                                
  7. (How about a couple of those sometimes-annoying [?] “think outside the box” BTs?  🙂 ). Forrest left home running. He ran a ways and then turned left, ran the same distance and turned left again, ran the same distance and turned left again. When he got home, there were two masked men. Who were they? The catcher and umpire. (Forrest was running the bases.)  Frank Green, Don Hayes, Marcia Morriset
  8. A man stands on one side of a river, his dog on the other. The man calls his dog, who immediately crosses the river without getting wet and without using a bridge or a boat. How did the dog do it? The river was iced over, and the dog crossed on the ice. (Others possible?)  No solvers.
  9. Write numerically the number ten thousand ten and ten thousandths.  10010.010  Frank Green, Alexis Avis, Anita Dixon.
  10. A four-digit (base 10) number has a 4, a 5,and two 8’s as its digits. What is the smallest such number that is divisible a) by 4? 4588  b) by 5? 4885  Frank Green, Alexis Avis, Anita Dixon.
  11. One million two hundred thirty four thousand five hundred sixty seven divided by 8 = x with a remainder of y.  Find y. 7  Frank Green, Alexis Avis, Anita Dixon.
  12. What is the largest prime factor of 87! + 88! ?    (NOTE: n! = 1*2*3*4* . . . .*n) 89.  (Nifty solution on request.)  Anita Dixon.
  13. You have 3 boxes, one red, one blue, and one green.  Inside each box is one marble which is the same color as the box.  Someone takes each ball out and puts in a different colored box.  You open only the red box and pull out a green marble.  What color is the marble in the blue box?  Red  Frank Green, Anita Dixon.
  14. A golf ball falls randomly onto a circular green of radius 10 meters, with the hole in the exact middle.  What is the probability that the ball falls within 1 meter of the cup?  1/10 (or equivalent)  Frank Green, Anita Dixon.
  15. What is the sum of the first three primes? 10  Frank Green, Anita Dixon.

BONUS 1:  (This puzzle was used last fall as NPR’s Puzzle Challenge on their “Sunday Puzzle” portion of the Weekend Edition program.  I found it while reading an article about one of the over 2000 folks who solved it.)  Think of a popular tourist attraction with two words.  The second, fourth, and sixth letters of the second word, in order, spell the first name of a famous author.  The last four letters of the first word spell the author’s last name.  Who is the author, and what is the tourist attraction?  Ayn Rand.  Grand Canyon  Frank Green, Alexis Avis, Anita Dixon.

Bonus 2:  Which number(s) from 1 to 100 inclusive has/have the most factors?  60, 72,  84, 90, and 96 all have twelve. Anita Dixon.

Bonus 3:  Take any prime number greater than 3, square it and subtract 1.  What is the largest number that must be a factor of the result.  24.  (this is a surprising result, with a beautiful and elegant ‘proof’.  Holler if you’d like to see it.) Anita Dixon.

Bonus 4:  See #6 above. The multiple choices there don’t seem to account for a male speaker.  But isn’t it possible?  What would the answer be if so?  (Let’s assume all relationships are biological [no step/half siblings, parents, etc.]  Son-in-law. Anita Dixon.

SUMMARY: Solutions & Solvers, Jan/Feb BTs

by | Feb 27, 2022 | Brain Teasers, Various Answers | 0 comments

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

  1. Simplify: (2 + 0 + 2 + 2)2 – (2 – 0 – 2 – 2)2 + 20 – 22   36 – 4 + 20 – 22 = 30.  Rita Barger, Amy Ragsdale, Anita Dixon, Jenni Wall
  2. When could it believably be said that 8 + 8 = 4? (Suspect there may be multiple answers?)  ONE ANSWER:  Eight hours added to 8:00 is 4:00.  (Another involved rolls of toilet paper – creative!) Rita Barger, Jenni Wall, Alexis Avis
  3. 9,811,438,761 divided by 9 leaves what remainder? 3  Rita Barger, Amy Ragsdale, Anita Dixon, Jenni Wall, Frank Green, Alexis Avis
  4. A board 2.5 meters long is divided into ten equal pieces. How long – in centimeters – is each piece? 25 (cm).  Rita Barger, Amy Ragsdale, Anita Dixon, Jenni Wall, Alexis Avis, Frank Green
  5. Two different prime numbers are selected at random from among the first ten primes. What is the probability that their sum is 24?  (Express your answer as a fraction.)  1/15.  (There are 45 different selections. 3 of them have a sum of 24. )  Rita Barger,  Anita Dixon, Frank Green
  6. Too easy? Merry Christmas!  😊   7.  Rita Barger, Amy Ragsdale, Anita Dixon, Frank Green, Alexis Avis, Jenni Wall
  7. Find the least common multiple of 10, 15, and 18. 90.  Rita Barger, Amy Ragsdale, Anita Dixon, Frank Green, Alexis Avis
  8. How many distinct rearrangements are there of the word “MATH”?  24. Rita Barger, Anita Dixon
  9. You have 628 cm. of string, which you may form into either a square or a circle. Which figure will yield the most area?  The circle.  Rita Barger, Amy Ragsdale, Anita Dixon, Alexis Avis
  10. The maximum speed of a zebra is 40 mph. IF the zebra could keep up that speed, how long would it take it to run one mile?  1.5 minutes. Rita Barger, Amy Ragsdale, Anita Dixon, Alexis Avis
  11. How many positive integers less than 124 are divisible by 2, 3, and 5?  4. (30, 60, 90, 120) Rita Barger, Anita Dixon, Alexis Avis
  12. Find the product of the first five even whole numbers. 0. (0 is the first even whole number.)  Rita Barger, Anita Dixon
  13. Which of the following best describes the GCD of two numbers?  a)  always even  b)  always odd  c)  never prime  d) none of these.  d.  Rita Barger, Anita Dixon,  Don Hayes, Frank Green
  14. There are several combinations of whole numbers whose sum is 12.  Find the pair with the greatest product. 6, 6. Rita Barger, Amy Ragsdale, Anita Dixon, Alexis Avis, Frank Green
  15. Find two examples of numbers that have exactly three factors (no more, no less).  The first 3:  4, 9, 25.  Rita Barger, Amy Ragsdale, Anita Dixon, Alexis Avis
  16. 10 Visual Brain Teasers Kids will Love! | Teach Starterc. Rita Barger,  Anita Dixon, Alexis Avis

 

Bonus 1:  See  #3 above.  What slick math tidbit makes this problem easy to solve without paper, pencil, or calculator?  The old technique of “casting out 9’s”. (Inquire for further details.)   Rita Barger, Amy Ragsdale, Anita Dixon, Alexis Avis

Bonus 2:  Square a two-digit number and subtract one.  Under what conditions will the result be prime?  Never.  (Viewed as the difference of two squares, it quickly factors.  So it has two factors [and neither can be 1. why?], so it can’t be prime.)  Don Hayes

Bonus 3:  Write the number (124 – 54) as a product of primes.  7*13*13*17  (the number is 20111, and one could find the answer by trial/error.  But it’s easier/luckier to recognize it as a “difference of two squares”, factor it (two steps), and work with smaller numbers.)  Rita Barger, Anita Dixon

Bonus 4:  See #15.  Do you know (or can you deduce) what is true in general about integers with exactly three factors?  They are always the squares of primes.  (They are not always odd, although 4 is the only exception. [why?]) Full or partial credit to Rita Barger, Amy Ragsdale, Anita Dixon, Alexis Avis

SUMMARY: Solutions & Solvers, Fall ’21

by | Dec 4, 2021 | Brain Teasers, Various Answers | 1 comment

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

  1. (This one is actually designed for the brand new subscribers who are Math Buddy volunteers (with young children) in Springfield, but of course anyone may answer!)  Find the next 3 entries in this sequence:  1, *, 2, **, 3, ***, 4, ____, ____, ____.  ****, 5, *****  Jim Waterman, Anita Dixon 
  2. What’s the smallest whole number that’s a multiple of 1,2,3,4,5, and 6? 60  Rita Barger, Anita Dixon
  3. Find the only pair of whole numbers whose product is one million, yet neither whole number has a zero in it. 64 and 15,625  Rita Barger, Anita Dixon
  4. (Another one for the Math Buddy volunteers (see #1) but I think this one might be fun and/or challenging for any of us!  For full credit, also list the numbers found. 1,2,3,4,6,8,9 (and one answer found a 7!? ) Rita Barger, Anita Dixon
  5.  What is the largest whole number such that 7 times it is still less than 1000? 142 Rita Barger, Anita Dixon
  6. List one number between 1.999 and 2.  If this is not possible, mark NP and tell why. 1.9991 (just one of an infinite number of correct answers.)  Rita Barger, Anita Dixon
  7. (A repeat?  One of my classic favorites!)  A man buys a horse for $60, sells it for $70, buys it back for $80, and sells it one last time for $80.  How much money, if any, did the man make on the series of transactions?  He made $10. Rita Barger, Anita Dixon I thought I had typed $90 for the very last amount (the usual way the problem is stated), but alas I didn’t, so the correct answer is $10 – as I posted it – and Anita & Rita caught it!!
  8. What whole number less than 50 has the largest number of factors? 48. (10 factors) Rita Barger, Anita Dixon
  9. Frank Farmer bought 2568 inches of fencing.  He immediately used five yards for a small animal pen.  How many feet of fencing were left after that? 199 ft. Rita Barger, Anita Dixon, Frank Green
  10. How many gallons of gasoline could be saved in one year by a fuel-efficient (non-electric :-)) car getting 32 miles per gallon over a gas-guzzler getting 14 mpg?  (Assume average yearly mileage is 9000 miles.)  Roughly 361 gallons. Rita Barger, Anita Dixon
  11. From a square of side 1, a new square is formed by connecting the midpoints of each side.  What is the area of the new square?  1/2 sq. units. Rita Barger, Anita Dixon
  12. A pizza company advertises that its 18-inch (diameter) party pizza has more pizza that two medium (12-inch) pizzas.  Are they right?  Yes.  The party pizza has area 81*pi sq in, while the two medium pizzas add up to 72*pi sq. in. Rita Barger, Anita Dixon
  13. All kinds of different interpretations here, believe it or not [ask for details, if you dare], so for a variety of reasons, I took anything close to 14, 15, 16, or 17.  🙂  Rita Barger, Anita Dixon
  14. Short.  Rita Barger, Anita Dixon, Don Hayes
  15. You have four colored chips – 2 black, 1 yellow, and 1 white.  The are in a horizontal line, left to right. The white chip is directly to the left of a black chip, and neither black chip is on an end.  How are the colored chips aligned? W, B, B, Y. Rita Barger, Anita Dixon
  16.  See #15.  Same set-up, but the chips are now 2 red, 1 yellow, and 1 green.  The yellow chip is not on an end, and the two ends are different colors.  The red chips are not adjacent, and the green chip is on the far right.  How are the chips aligned now?  R, Y, R, G  Rita Barger, Anita Dixon
  17. (These last two are also repeats, AND they are added in honor of the MSTA subscribers!)  Find the largest fraction that a) has a denominator of 17, and b) when added to 1/3, keeps a sum less than 1.  Technically the answer is 11/17, but I also took 11, with misplaced focus on numerator.  Rita Barger, Anita Dixon
  18. If a certain book is 4th from the left on a bookshelf, and also 6th from the right on the same shelf, how many books are on the shelf?  9 Rita Barger, Anita Dixon

Bonus #1  (a repeat).  If you were spelling out the whole numbers, how far would you have to go before you first used the letter a ?    One thousAnd.     Linda Ward, Frank Green, Rita Barger, Jim Waterman, Anita Dixon  (This may be one of the rare – if not only – time(s)  a Bonus answer has more correct solutions than any of the BTs !)

Bonus #2  See #4 above.  Can you design something similar to above (or entirely different if you choose) that has more hidden numbers than the figure above does?  (I believe the hidden conditions is that they all must be connected.)  No brave takers.

Bonus #3  See #8 above.  Can you find the smallest whole number with exactly 10 factors? 48.  Frank Green, Rita Barger, Anita Dixon

Bonus #4:  See # 11 above.  What if the original square has side x ? (x^2)/2 Rita Barger, Anita Dixon

Bonus #5:  Arrange these four numbers from smallest to largest.  (If any are equal, put = signs between them)                                                                                              π, 3.14, 22/7, 3.1416    3.14, π, 3.1416, 22/7  Rita Barger, Anita Dixon

 

SUMMARY: Solutions & Solvers, Jul/Aug ’21 BTs

by | Sep 2, 2021 | Brain Teasers, Various Answers | 0 comments

REMINDERS:  Answers in red.  Solvers (submitted/correct) in blue. (Forgive any omissions, but feel free to inform.) Comments in green. For further elaboration on solutions, please feel free to ask! Consider the complement of any acute angle and the supplement of any obtuse angle.  Is one of those always bigger (which one?) or might they vary?

  1. Consider the complement of any acute angle and the supplement of any obtuse angle.  Is one of those always bigger (which one?) or might they vary? The answer could easily vary, depending on the angles chosen.  Rita Barger, Anita Dixon, Alexis Avis.
  2. Consider any positive rational number and that number’s square root. .  Is one of those always bigger (which one?) or might they vary? The answer could easily vary, depending on the number chosen.  (Square roots between 0 & 1 are bigger than the original number.)  Rita Barger, Anita Dixon, Alexis Avis.
  3. Your father.  Rita Barger, Anita Dixon, Alexis Avis, Dan Felshin, Pat Campbell
  4. Which two whole numbers with a sum of 50 will give the largest product?  Accepted answers of (25 & 25) or (24 & 26), depending on whether you assumed the numbers were distinct).  Rita Barger, Anita Dixon, Alexis Avis, Frank Green.
  5. Two (distinct) numbers are chosen at random from the set (2,3,5,7) and multiplied together.  How many of the possible products are odd?  Three of the (six) possible products are odd. Rita Barger, Anita Dixon, Frank Green
  6. Find the product: (1 – 1/2) * (1 – 1/3) * (1 – 1/4) * . . .  * (1 – 1/10)  1/10.  Anita Dixon
  7. Find the product of all the even integers between -25 and 25. The product is 0 (since 0 is an even integer between -25 and 25).  Rita Barger, Anita Dixon, Frank Green.
  8. If the radius of a circle is tripled, how much is the area of the circle increased?  The area is increased ninefold.  Rita Barger, Anita Dixon, Alexis Avis.
  9. In a strange house, there are 6 rugs and on each rug sits 6 six-legged tables.  On each table, there are 6 six-legged creatures.  What is the total number of creature legs in this silly story? 1296.  Rita Barger, Anita Dixon, Alexis Avis.
  10. The mean of all the primes between 40 and 50 is subtracted from the median of the same primes.  To two decimal places, what is the absolute value of the answer?  0.67  (The mean is 43.67, the median is 43.) Rita Barger, Anita Dixon, Frank Green.
  11. HELP!!  I thought certain I had added at least one BONUS to the Brain Teasers this time around, but either I never did, or it has disappeared in various revisions.  CAN YOU HELP ME?  Help me solve this riddle (send me the Bonus or let me know it was never there) – and receive credit for this Brain Teaser!!  Apparently I did NOT include a Bonus in this group.  (A first perhaps?)  Anita Dixon, Alexis Avis.
  12. Have you seen the US Post Office’s new stamp??!  Check in out here.  Imagine – a puzzle on a stamp!!  Decipher the message, and get your BT credit! (you can look it up of course, but it’ll be more fun if you don’t.)  More Than Meets the Eye.  (Check out the link.  The answer was cleverly hidden in the caption. 🙂 ).  Frank Green
  13.  I don’t buy this ‘Only for Genius’ stuff.  This one’s not hard – what goes where the ? is?? I took either 25 or 1 (depending on whether your viewed the ? as a product or a factor. Anita Dixon, Alexis Avis, Frank Green.
  14. The number 555,555 can be expressed as the product of two 3-digit numbers.  What is the smaller of those two? 715. Anita Dixon.
  15. Two standard dice are rolled.  What’s the probability (in fraction form) that the sum of the two numbers is prime? 15/36 (= 5/12). Frank Green. Alexis Avis

SUMMARY: Solutions & Solvers – May/June 21

by | Jul 9, 2021 | Brain Teasers, Various Answers | 0 comments

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

  1. This brain teaser first appeared on 5/3/21.  This is almost a ‘prime countdown day’.  a)  Why is it NOT?  b) When will the next true prime countdown day occur?  a) 1 is not a prime.  b) 7/5/32.  Rita Barger, Amy Ragsdale, Anita Dixon, Alexis Avis, Jim Waterman
  2. What has a face and two hands, but no arms or legs?  A [non-digital] clock (or watch) face.  Rita Barger, Amy Ragsdale, Anita Dixon, Alexis Avis, Jim Waterman, Frank Green, Marcia Morriset
  3. True or False? The sum of the reciprocals of the factors of 4 is greater than 2. False.  It adds to 1.75 (or 7/4, etc.) Rita Barger, Amy Ragsdale, Anita Dixon, Alexis Avis, Jim Waterman.
  4. You are running a marathon and overtake the person in 2nd What place are you in now? 2nd. Rita Barger, Amy Ragsdale, Anita Dixon, Alexis Avis,  Frank Green, Marcia Morriset
  5. One-fourth is one-third of what number? 3/4 Rita Barger, Amy Ragsdale, Anita Dixon, Alexis Avis, Jim Waterman
  6. If 3x + 1 is an even number, what is the next larger even number?  3x + 3  Rita Barger, Amy Ragsdale, Anita Dixon, Jim Waterman, Frank Green
  7. Math Brain Teaser for Middle School Students D.  (10).  Rita Barger, Amy Ragsdale, Anita Dixon, Jim Waterman
  8. If p is an odd prime, find the sum of all the factors of 4p. 7p + 7  Rita Barger, Amy Ragsdale, Anita Dixon,
  9. SIMPLIFY:  7/22 ÷ 14/44  1.  (7/22 = 14/44) Rita Barger, Amy Ragsdale, Anita Dixon, Alexis Avis, Jim Waterman, Frank Green, Marcia Morriset
  10. I thought of beekkeeping (or -er), but most had bookkeeper (-ing).  All would work, of course. Rita Barger, Amy Ragsdale, Anita Dixon, Alexis Avis, Jim Waterman, Frank Green
  11. More English oddities:  ‘FACETIOUS’ is the only word in English which has this property.  Can you spot it (the property)?The word contains all 5 vowels, and in order.  (Jim Waterman pointed out that facetiously ALSO includes the y!) Rita Barger, Amy Ragsdale, Anita Dixon, Alexis Avis, Jim Waterman, Frank Green
  12. The radius of Circle A and the diameter of Circle B are both 4 cm.  What is the ratio of the smaller circle’s area to the larger’s? 1:4 Rita Barger, Amy Ragsdale, Anita Dixon, Alexis Avis, Jim Waterman, Frank Green
  13. Suppose John’s average heart rate is 72 beats/minute.  Given that, how many times would his heart beat in a day? 103,680. Rita Barger, Amy Ragsdale, Anita Dixon
  14. Pick ANY 6 negative integers.  Which is bigger – their sum or their product, or might it vary? The product, always.  (With six negatives, the product will always be positive, the sum negative.) Rita Barger, Amy Ragsdale, Anita Dixon, Alexis Avis
  15. What is ¼ of 240?  (Keeping answer in exponent form is certainly acceptable – even advisable!) 238  Rita Barger, Amy Ragsdale, Anita Dixon, Alexis Avis.

BONUS 1:  See #1.  There’s another fairly famous math entity we could connect in a ‘countdown’ to  5/3/21.  What is it, and when will the next one occur?  (Possible multiple answers?)  The digits are all Fibonacci numbers.  Next one would be 8/5/32.   Rita Barger, Anita Dixon

BONUS 2:  A circle of radius 8 is cut into a large plywood board.  Find the length of the side of the largest cube that will pass through the hole. 8*(SQRT2)  Amy Ragsdale, Anita Dixon, Jim Waterman 

BONUS 3:  See # 6.  What are the next two odd numbers after the original 3x + 1?  3x +2, 3x + 4.  Rita Barger,  Amy Ragsdale, Jim Waterman 

BONUS 4:  If x*y = the product of x and y, and x#y = x – y, then find [2(8#12)] # [(3*2)#5]  – 9. (-8 # 1 = -8 – 1 = -9) Rita Barger, Amy Ragsdale, Anita Dixon

Possible Zoom Topics

by | May 16, 2021 | AfterMath Adventures - Fun/Learning, Various Answers | 0 comments

Possible Fun One-Shot Zoom Topics

Designed to be fun, and to fit in your hectic schedules, here are some possible topics that could surface at AME in the near future. All of these would be short (30-40 minutes), with time for questions. 

Interesting & Weird Mathematicians – And Stories About Them!  A standing ovation for a talk where no words were spoken?!  The ultimate interesting and WEIRD mathematician!  What there is no Nobel Prize in math.  Napier’s psychic rooster?!  And many, many more!!   Likely a Part 1 and a Part 2 (at least?)

Lewis & Clark on the Missouri River  A quick summary – with pictures of the places – of some of the Corp of Discovery’s adventures (good and bad) on the Missouri River part of their 3-year journey.

Spitballs From the Back Row  A collection of stories and anecdotes from in/near the classroom (including some of my own mishaps!), with some brief discussion of how we can learn from them.  Likely a Part 1 and a Part 2.

Brief ‘Snapshot Travelogues’  If you like the Photo of the Week in the Mailings, you should like this adventure.  We’ll focus on a country, OR a region of the US, or a similar attribute, and focus there for awhile.  Likely a dozen or more possibilities here!

And others are on the ‘drawing board’ at this very moment!

I’d LOVE to hear if anything above ‘grabs you’, as well as any preferences, suggestions (day of week, time of day, other topics?).  EITHER leave a COMMENT below, or use the CONTACT  FORM ,or simply REPLY to the Mailing.  Input always appreciated!