Summary: Solutions & Solvers – Fall ’19 BTs

Fall ’19 Brain Teasers – SUMMARY

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. I recently cleaned out my MSU ‘Emeritus’ office (whew!).  I found two items tacked to a bulletin board. One was Hampton’s Travel Tips #628. The other was a coupon for a drawing, and was numbered 9998896.  Knowing my fascination with numbers, why might I have saved each of these?  (LOTS of possible answers here, of course, and most will receive credit.)  628 is formed from the first 2 perfect numbers (6 & 28).  9998896 can be turned upside down and still reads as a number. (To this day, I’m still not SURE which was the ticket number!)  One or both parts: Rita Barger.                       
2. Perhaps you’ll remember KrazyPic1 from Aug 26.  Assuming all 4 ‘highlighted’ digits there are used once-each in a 4-number code, how many possible codes are possible? 24  Rita Barger, Amy Ragsdale.
3. Name two consecutive prime numbers whose product is 899. 29 & 31.  Rita Barger, Amy Ragsdale, Don Hayes.
4. What starts today, can’t be found at noon, and is required to end sunset?  The intended answer is ‘the letter t’, but Rita Barger submitted ‘darkness’, which I LOVE!!  Rita Barger, Amy Ragsdale, Don Hayes, Frank Green.
5. Melissa went to dinner with Andrew, George, and Ulysses.  She ate along, but they all showed up to pay for the meal.  Why? Andrew Jackson, George Washington, and Ulysses S. Grant are faces on the bills she used to pay for the meal. Rita Barger, Amy Ragsdale, Frank Green.
6. Which number is the third smallest of these?  0.3, 3.03, 0.303, 3.3303, 3.303  Rita Barger, Amy Ragsdale, Frank Green.
7. Find the sum of the digits of the largest even 5-digit number that is not changed when its ones and ten-thousands digits are interchanged. 43 (for 89998) Frank Green. Partial credit – Amy Ragsdale.
8. If there are four cars ahead of a car, four cars behind a car, and a car in the middle, what is the fewest number of cars in the line? 5 Frank Green, Amy Ragsdale.
9. (Is this a repeat?) What three letters can be arranged to describe a beverage, a verb, and a homonym? Tea, eat, ate. (Perhaps that should read homophone?) Frank Green, Amy Ragsdale, Don Hayes. 
10. 1/5 ÷ 1/5 ÷ 1/5 ÷ 1/5 = ? 25  Frank Green, Amy Ragsdale.
11. Find a palindrome that is a cubic number less than 2000. 1331 = 11^3.  Amy Ragsdale, Don Hayes.  (Amy R notes I provided a clue by making this problem #11.  :-).  Was that intentional, she wonders.  I’ll never tell.)
12. How can you plant ten seeds in eight rows with three seeds in each row? Plant them in a 3 x 3 grid, and using all directions, there will be eight rows of three. No correct submissions.
13. (Repeat) Using each of the ten digits once, find two five-digit numbers with the greatest possible product. Several close answers here.  Amy Ragsdale’s 97530 x 86421 = 8,428,640,130 is largest received, but not officially verified as ‘largest’.
14. (Repeat) What’s a good method to quickly solve this problem mentally?   (2019 + 2019) x 50  It’s 2*2019*50.  Group the 2*50 to get 100 and 100*2019 = 20190. Amy Ragsdale.
15. (Repeat) What part of a half square foot is a half foot square?  A half ‘square foot’ is 0.5 ft^2.  A ‘half foot’ squared (or a square of side half-foot) is 0.5*0.5, or 0.25 ft^2.  So the latter is half the former in size.  No correct submissions.
16. (Repeat) How many cards must you draw from a deck of 52 cards to be sure that at least two are from the same suit? 5.  No correct submissions.

BONUS 1:  We’ve done several variations of the billiard ball/counterfeit coin problem recently.  Here’s a most interesting new one (at least to me):  You have the usual 12 counterfeit coins, and now you know that TWO of them are counterfeit and are too heavy (but both ‘bad’ coins weigh the same.).  Can you determine/identify BOTH the bad coins in five (or less) weighings with the same balance scale? Yes, it can be done.  Details provided upon request.  Don Hayes

BONUS 2: For real math nerds!  🙂  How many pairs of integers p and q exist such that (p^2+1)/q and
(q^2+1)/p are both integers? (Shared by Don Hayes, who says he saw it in a movie.)  Don Hayes and Amy Ragsdale each provided or have solutions.  Details upon request.

BONUS 3:  Arrange the integers from 1 – 16 in such a way that the sum of any two consecutive integers is also a square. 8, 1, 15, 10, 6, 3, 13, 12, 4, 5, 11, 14, 2, 7, 9, 16.  Amy Ragsdale.  (Other than reversing direction, I know of no other ways this can be done.)

BONUS 4:  See #12 above.  The source I used for this problem says “ten seeds in ten rows”.  I’m not sure I believe it?!  (I can get 8 rows, so I changed wording, but . . .).  Can anyone help out?

CREATIVITY BONUS A:    The Herman cartoon below has had the caption (temporarily) removed.  Submit your own caption and get us laughing!!  (Later we may decide we like some of these submissions better than the original!)

CREATIVITY BONUS B:  Submit a good Creativity Bonus idea for this section!!