SUMMARY: Solutions & Solvers, Mar/Apr ’21 BTs

by Larry Campbell | Apr 30, 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, please feel free to ask! 

1. Consider π, 3.14, and 22/7. Write them in order from smallest to largest, or, if any are equal, mark EQUAL in appropriate spots..3.14, π, 22/7.  Anita Dixon, Frank Green, Rita Barger, Amy Ragsdale, Jonah Albertson.
2. Are there any days this year (’21) where the day# times the month# = the year# [21]? Yes.   Jan 21, Mar 7, and July 3. Anita Dixon, Frank Green, Rita Barger, Amy Ragsdale, Jonah Albertson, Alexis Avis
3. When if ever, is 1/x  bigger than x? When x is between 0 & 1, or when x is less than -1.  Anita Dixon,  Jonah Albertson, Rita Barger.  Partial Credit:  Amy Ragsdale
4. When, if ever, is a number less than its square?  When the number is less than 0 or greater than 1.  Anita Dixon, Rita Barger, Jonah Albertson. Partial Credit:  Alexis Avis and Amy Ragsdale
5. See picture and question -> Accepted 6 or 8, depending on whether submitter thought the holes in front went all the way through (and counted them separately.)   Anita Dixon, Frank Green, Rita Barger, Amy Ragsdale, Alexis Avis
   
6. A cubic foot of air weighs 0.0765 pounds. What is the weight of the air in a 20 x 30 x 10 foot classroom? 459 lbs. Anita Dixon, Frank Green, Rita Barger, Amy Ragsdale, Alexis Avis
7. Halley’s Comet appeared in 1758, 1834, 1910, and 1986. When will it next appear? 2062.  (Appears every 76 years.) Anita Dixon, Frank Green, Rita Barger, Amy Ragsdale, Alexis Avis
8. If grasshoppers can jump 59 times their length, how far can a 1.5 inch grasshopper jump? 88.5 in. Anita Dixon, Frank Green, Rita Barger, Amy Ragsdale, Alexis Avis
9. This past Saturday was 4/3/21. Let’s call that a ‘countdown day’.  When will the next ‘countdown day’ happen?  May 4, 2032. [5/4/32) (Other clever variations also suggested.)  Anita Dixon, Frank Green, Rita Barger, Amy Ragsdale, Alexis Avis
10. A penny weighs approximately 0.1 ounce. Using that figure, how much would a ton of pennies be worth? $3,200. Anita Dixon, Frank Green, Rita Barger, Amy Ragsdale, Alexis Avis
11. What is the perimeter of a square that has the same area as a 4 cm x 9 cm rectangle? 24 cm. Anita Dixon, Frank Green, Rita Barger, Amy Ragsdale, Alexis Avis
12.  C. (25) (in this case: in order, from left to right.) Anita Dixon, Rita Barger, Amy Ragsdale, Alexis Avis
13. There are (at least) two 2-digit numbers that are equal to 7 times the sum of their digits. Find at least one of them.. 21, 42, 63, 84. Anita Dixon, Rita Barger, Amy Ragsdale
14. Write the numeral for the number eight billion, forty thousand, two 8,000,040,002 Anita Dixon, Rita Barger, Amy Ragsdale, Alexis AvisBONUS 1 See # 13 above.  Can you  find more than two such numbers?  See above for others.  Amy Ragsdale, Anita Dixon. 

 BONUS 2  Note that it takes 6 syllables to pronounce Kansas City Royals. but only 5 to pronounce St. Louis Cardinals.  On the other hand pronouncing either New York Yankees or Boston Red Sox only takes 4 syllables.  Lets take Major League Baseball, and add NFL football teams for fun, as well.  What are the extremes on syllables for City and Mascot?  Hurriedly, I can think of one team that takes 7 syllables, and one team that only takes 3.  Can you find them?  Can you find teams with more than 7? Or less than 3? Teams found with 3 syllables:  New York Mets, New York Jets, Cleveland Browns, Tampa Rays(may not be full name?)  Teams with 7 syllables:  Arizona Diamondbacks, Philidelphia Eagles, Philadelphia Phillies, Indianapolis Colts.  Teams with 8 syllables:  San Francisco 49ers. Contributers: Amy Ragsdale, Frank Green, Alexis Avis, Steve & Bobbi Kneeshaw.

BONUS 3 Did you see Dean Martin’s Fun Fact involving 80 beers and your birth year?  🙂  Can you explain how it works? [See also note at link for more precise updates.]  Ask for details.  Amy Ragsdale, Anita Dixon.

BONUS 4  See #10.  How much would those pennies weigh?  (Be alert!)  A ton of pennies would weigh             a ton.  🙂 Amy Ragsdale, Anita Dixon.

SUMMARY: Solutions & Solvers; Jan/Feb ’21 BTs

by Larry Campbell | Mar 4, 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, please feel free to ask! 

1. Why are some folks already looking forward to Jan 1, ’23 at 5:08:13?  The date lists the first 7 digits of the Fibonacci Sequence. (1,1,2,3,5,8,13,21,34, . . .) Rita Barger, Amy Ragsdale.
2. What is an interesting feature of the three words job, polish, and herb? (Yes, probably lots of ‘correct’ answers.) They each have different meanings when capitalized. (Similar versions accepted.) Rita Barger, Amy Ragsdale, Frank Green.
3. Find a number less than 100 that is increased by one-fifth of its value when its digits are reversed. 45 Rita Barger, Amy Ragsdale, Frank Green, Dan Felshin.
4. You close your book, forgetting to put in a bookmark, but you remember (who knows why?) that the product of the pages is 16002. What were the page numbers? 126 & 127  Rita Barger, Amy Ragsdale.
5. A square plot of 160 acres is surrounded by a fence of total length 2 miles. How large a square plot will a fence of four miles surround?  640 acres.  (Doubling one side of a square quadruples the area.)   Rita Barger, Amy Ragsdale, Don Hayes.
6. In what order are the following digits? 0, 2, 3, 6, 7, 1, 9, 4, 5,  In reverse alphabetical order, considering the digits’ spellings.  Rita Barger, Amy Ragsdale.
7. Al, Joe, Carl, and Dave were standing in line. Dave was not first, Joe was between Al and Carl.  Al was between Dave and Joe.  In what order were they standing? Carl, Joe, Al Dave. Rita Barger, Amy Ragsdale, Frank Green, Dan Felshin
8. This BT is stated as a question, so let’s go with it. If you think the answer is YES, solve it.  If you think it is NO, tell why. NO, It cannot be solved.  Briefly, the sum of 3 odd numbers is never even. Rita Barger, Amy Ragsdale, Frank Green, Don Hayes, Dan Felshin
9. 1 + 1 + 1 x 0 + 1 + 1 = ____4____ Rita Barger, Amy Ragsdale, Frank Green.
10. 10.You have $6.60 in dimes and quarters (only).  If there are 36 coins how many quarters to you have? 16 dimes and 20 quarters. Rita Barger, Amy Ragsdale, Frank Green.
11. What is exactly (in fraction form) the sum of ½ and ¼ ? 3/4. Rita Barger, Amy Ragsdale, Frank Green.
12. Your “I Love Math” club contains 6 men and 4 women. How many different committees of 2 men and 3 women are possible?  60.  Amy Ragsdale.
13. How many squares (of any size) are in the drawing below?  40  Rita Barger, Amy Ragsdale.
    
14. X is Y’s brother, but Y is not X’s brother. Who is Y?  X’s sister!  Rita Barger, Amy Ragsdale, Frank Green.
15. How many cards must you draw from a standard 52-card deck to be sure that at least two are from the same suit? 5. Rita Barger, Amy Ragsdale
16. The missing digit is 8.  (These are the 1st 12 digits in the decimal representation of PI.) Rita Barger, Amy Ragsdale

BONUS 1:  See #13 above.  How many rectangles are in the picture.  (I don’t know this answer!  It just came up in the Fun With Math Class. I’d LOVE to hear your conjectures.[I’ve heard one already.])  118??  I’ve got question marks because this is a relatively new problem to me (proposed by Kurt Killion), so I don’t have an official answers.  HOWEVER, both Kurt Killion and Amy Ragsdale have come up with 118 now.  And, I’ve seen Kurt’s work and can’t find any flaws.  I may carry this over to next time to give others time to confirm 118, or come up with another answer (along with reasoning.)

BONUS 2:  See #15 above.  How many would you have to draw to make sure you had at least 2 hearts? 41.  Rita Barger, Amy Ragsdale

SUMMARY: Solutions & Solvers, Jul/Aug 20 BTs

by Larry Campbell | Sep 4, 2020 | Brain Teasers, 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, 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.

SUMMARY: Solutions & Solvers – May/June ’20 BTs

by Larry Campbell | Jul 11, 2020 | Brain Teasers, 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, please feel free to ask! 

1. What kind of cheese is made backwards? 🙂  Edam.  Frank Green, Rita Barger, Amy Ragsdale, Barbara Blue
2. What word begins and ends with ‘e’, but only has one letter? Envelope.  Frank Green, Amy Ragsdale, Barbara Blue, Don Hayes, Jim Waterman
3. The integer 1234 is not divisible by 11. By re-arranging the digits, can you find one or more integers that are divisible by 11?  (A yes/no answer does not get full credit. 🙂 ). Yes, there are several.  1243 is one, for example. Frank Green, Rita Barger, Amy Ragsdale, Barbara Blue, Don Hayes, Jim Waterman
4. A new-born infant was given 1 trillion dollars at birth.  (Wow!).  How much would he/she have to spend (average) each day to use all the money in 80 years (assuming no interest accrued)? Over 34 million!  ($34,223,134.84, to be exact.) Frank Green, Amy Ragsdale, Barbara Blue, Jim Waterman
5. How many spherical meatballs of radius 1 can you make from a spherical meatball of radius 3?  Exactly 27. Rita Barger, Amy Ragsdale, Barbara Blue,  Jim Waterman
6. (Repeat) The sum of A & B is 112.  A is 4 more than C, and B is 2 less than C.  What does C equal? 55  Frank Green, Amy Ragsdale, Barbara Blue
7. I am < 25.  My ones digit is twice my tens digit and the digits add to an even number.  What am I? 24  Frank Green, Amy Ragsdale, Barbara Blue
8. Find the area of the shaded region to the right.  –> 400 – 100*PI (sq cm). (Or roughly 85.9, if you prefer) Frank Green, Amy Ragsdale.
9. Take any 3 consecutive integers.  Square the middle one and multiply the outer two together.  Try this several times and see what you notice.  Can you formulate a rule? The middle number squared is always one more than the product of the other two.  Frank Green, Amy Ragsdale, Don Hayes
10. Peggy is writing down the integers from 1 to 1000. (Who knows?) She stops to rest after writing 630 digits. What is the last integer she wrote? 246  Amy Ragsdale
11. I am an odd integer.  If you take away one letter from my name, I become even.  What am I?  SEVEN.      Frank Green, Amy Ragsdale
12. When is it possible to subtract 1 from 19 and get 20?  When is half of 13 = 8?  With Roman numerals. Take one from IXX and you have XX.  Cut XIII in half (horizontally) and you have VIII left.      Amy Ragsdale
13. How many prime factors does 2020 have, and what are they? 3 prime factors.  (2, 5, and 101)  Frank Green, Amy Ragsdale
14. How many two-digit integers are divisible by either 3 or 5?  42        Frank Green
15. Find 1423² – 1418². (You can probably crank this out fairly easily [?], but there’s a clever way to do it very quickly.)   14205   Frank Green, Amy Ragsdale
16. What is the closest positive number (not just integer) to 0? There is NO such ‘closest’ number to 0. (Isn’t math amazing?!) Amy Ragsdale, Partial credit – Frank Green, 

BONUS 1:  Ten ping-pong balls are numbered from 1 to 10.  If two balls are drawn at random, what is the most likely sum of the numbers on the balls? (this can be done with knowing any probability.)  11. (Five different possible ways, more than any other sum.) Frank Green, Amy Ragsdale, Barb Blue

BONUS 2:  See #8 above. Re-label the 20 cm distance to a more general x cm.  Find an expression for the area of the shaded region. (3/4)*x^2*PI (sq cm) Frank Green, Amy Ragsdale

BONUS 3:  Every red card at an auto show was a sports car.  Half of all blue cards were sports cars.  Half of all sports cars were red.  There were 44 blue cars and 30 red cars.  How many sports cars were neither red nor blue. 8   Frank Green

BONUS 4:  Can you find a ‘fraction’ (one integer/another) whose decimal equivalent is the (ever-repeating) .3434343434  . . . ?  34/99  Amy Ragsdale

BONUS 5:  See #9 above.  If you have conjectured a ‘rule’ here, can you prove it?  One approach: Let the 3 numbers be x-1, x, x+1.  Then x^2 – 1 = (x+1)*(x-1).  Amy Ragsdale, Don Hayes

BONUS 6:  See #13 above.  How many total factors does 2020 have?  (and what are they?  That’s not a trivial question.  There’s actually a way to know how many w/out figuring them and counting directly.  Isn’t math wonderful? 🙂 ). There are 12 total factors.  (1,2,4,5,10,20,101,202,404,505,1010, and 2020)      Frank Green, Amy Ragsdale   

SUMMARY: Solutions & Solvers – Mar/Apr ’20 BTs

by Larry Campbell | May 2, 2020 | 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, please feel free to ask! 

1. (An old classic!)  Your unorganized sock drawer contains 18 white socks and 18 blue socks. How many times do you need to reach inside the drawer and take out a sock at random to guarantee you have a sock of each color? To be sure of a sock of each color, you’d have to pull out 19 socks.  Frank Green, Rita Barger, Jim Waterman, Amy Ragsdale
2. A cowboy rode into town on Friday. He stayed in town for three days and rode back out on Friday. How is this possible?  The horse’s name was Friday. 🙂  (Other alternate solutions here, including a perfectly-good-but-convoluted one from our ‘alternate solution master’ Jim Waterman.) Frank Green, Rita Barger, Jim Waterman, Jennifer Steele DeWeerdt, Amy Ragsdale
3. A palindrome is a number that reads the same forwards and backwards (like 202 or 454).  How many palindromes are there between 100 & 1000? 90.  Frank Green, Rita Barger, Jennifer Steele DeWeerdt, Amy Ragsdale
4. (A future hypothetical twist to a BT from last time)  The 52nd and 54th Presidents of the US have the same mother and father, but are not brothers and are not the same man.  How can this be? They are either brother/sister, or two sisters. Frank Green, Rita Barger, Jim Waterman, Jennifer Steele DeWeerdt, Amy Ragsdale
5. A woman was in her hotel room when there was a knock on the door. When she opened the door slightly, there was a man she’d never seen before.  He said, “I’m sorry, I have made a mistake, I thought this was my room” and walked away.  The woman went back into her room and phoned security.  Why was the woman so suspicious? No ‘official’ answer provided where I found this, but seems to me that if you’re going to your own room, you’d have a key and wouldn’t be knocking! Jennifer Steele DeWeerdt, Amy Ragsdale
6. a) How many ways are there to re-arrange the letters in the word ‘STOP’ ? 24 b) How many of those form a common-usage word? 6 (stop, spot, post, pots, tops, opts). Credit for 5 or 6 (OPTS is often omitted.) Frank Green, Rita Barger, Jennifer Steele DeWeerdt, Amy Ragsdale
7. Augustus De Morgan was a mathematician who lived in the 19^th century.  He once wrote, I was x years old in the year x².  a)  How old was he when he said that? Technically, he could have said it (past tense) in any year after 1849.  (See below).   B) What was his birth year?  1806 or 1807.  (43×43 = 1849, and it’s the only such year in the 1800s, so 1849 – 43 = 1806.)  Frank Green, Rita Barger, Amy Ragsdale
8. Several years ago, I drove a nail into a tree exactly five feet above the ground.  The tree grows at the rate of one half-foot a year.  Eleven years later, I returned to the tree.  How far above the ground was the nail? 5 feet.  (Trees grow from the top. :-))  Rita Barger, Jennifer Steele DeWeerdt
9. A man wanted to encrypt his password but he needed to do it in a way so that he could remember it. He had to use seven characters consisting of letters and numbers only (no symbols like ! or <). In order to remember it, he wrote down “You force heaven to be empty.” What is his password? U472BMT.   Amy Ragsdale (Partial credit to Frank Green for U4cH2BE.)
10. Guess the next three letters in the sequence:  GTNTL.ITS.  (GuessTheNextThree . . .) Frank Green
11. (Repeat) A digital clock forms palindromic times 114 times each day.  What is A) the least (2 min, between 9:59 and 10:01)  and B) the most amount of time between two palindromic clock numbers (1 hour, 10 min, twice, between 10:01 and 11:11 and then to 12:21)  Amy Ragsdale
12. (Ignore numbering snafu.  Can’t fix.)  Answer for #12 below:  They’re probably all stupid :-), but the most so is Boy #4, who will fall as a result of his own cutting.   Amy Ragsdale
13. 

BONUS1:  There are three light switches outside of a room:  #1, #2, and #3- They are connected to three light bulbs inside the room.  The door to the room is closed and you can’t see in. All three switches are currently off.  You need to figure out which switch belongs to which bulb. You can use the switches however you want to, but can only enter the room once. How do you do it?  Turn on Switch #1 (say) and leave on.  Turn on Switch #2 for ten minutes and then turn off.  Then go into room.  The bulb that is on belongs to Switch 1, the bulb that is warm belongs to Switch 2, and the other to # 3.  Frank Green.

BONUS 2: The Jan/Feb BTs mention Year Product Days  (For reminder, see Item 4 of this link.)  Are there any YPDs in each century with seven (7) or more YPDs?  xx24 each century has 7.  (1/24, 2/12, 3/8, 4/6, 6/4, 8/3, and 12/2.)  Frank Green.

BONUS 3:  A golf ball falls randomly onto a circular green 10 meters in radius, with the cup at the center.  What is the probability that the ball is within 1 meter of the cup?  Assuming A) the ‘falling’ is really random, and B) we’re talking ‘within 1 meter of the center of the cup’ (which, BTW, allows for a ‘hole in one’), then the answer is 1/100.  (Area of green = PI x 10 x 10 = 100(pi).  Area of inner circle = PI x 1 x 1 = (pi).)  Amy Ragsdale