March/April ’22 Brain Teasers

by Larry Campbell | Mar 3, 2022 | Brain Teasers | 4 comments

NOTE:  Newest BTs in red, Bonuses in blue, comments in green, updates in purple.

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.)
2. One half of N plus one half of N equals 5. Find the value of N.
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.  
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? 
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?
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.)
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?
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?
9. Write numerically the number ten thousand ten and ten thousandths.
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?  b) by 5?
11. One million two hundred thirty four thousand five hundred sixty seven divided by 8 = x with a remainder of y.  Find y.
12. What is the largest prime factor of 87! + 88! ?    (NOTE: n! = 1*2*3*4* . . . .*n)
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?
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?
15. What is the sum of the first three primes?

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?

Bonus 2:  Which number(s) from 1 to 100 inclusive has/have the most factors?

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.

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.]

Brain Teasers – Jan/Feb 2022

by Larry Campbell | Dec 10, 2021 | Brain Teasers | 2 comments

NOTE:  Newest BTs in red, Bonuses in blue, comments in green, updates in purple.

1. Simplify: (2 + 0 + 2 + 2)² – (2 – 0 – 2 – 2)² + 20 – 22
2. When could it believably be said that 8 + 8 = 4? (Suspect there may be multiple answers?)
3. 9,811,438,761 divided by 9 leaves what remainder?
4. A board 2.5 meters long is divided into ten equal pieces. How long – in centimeters – is each piece?
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.)
6. Too easy? Merry Christmas!  😊  
7. Find the least common multiple of 10, 15, and 18.
8. How many distinct rearrangements are there of the word “MATH”?
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?
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?
11. How many positive integers less than 124 are divisible by 2, 3, and 5?
12. Find the product of the first five even whole numbers.
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.
14. There are several combinations of whole numbers whose sum is 12.  Find the pair with the greatest product.
15. Find two examples of numbers that have exactly three factors (no more, no less).
16. 

Bonus 1:  See  #3 above.  What slick math tidbit makes this problem easy to solve without paper, pencil, or calculator?

Bonus 2:  Square a two-digit number and subtract one.  Under what conditions will the result be prime?

Bonus 3:  Write the number (12⁴ – 5⁴) as a product of primes.

Bonus 4:  See #15.  Do you know (or can you deduce) what is true in general about integers with exactly three factors?

Brain Teasers – Sep/Oct/Nov ’21

by Larry Campbell | Sep 2, 2021 | Brain Teasers | 2 comments

NOTE:  Newest BTs in red, Bonuses in blue, comments in green, updates in purple.

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, ____, ____, ____.
2. What’s the smallest whole number that’s a multiple of 1,2,3,4,5, and 6?
3. Find the only pair of whole numbers whose product is one million, yet neither whole number has a zero in it.
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.
5.  What is the largest whole number such that 7 times it is still less than 1000?
6. List one number between 1.999 and 2.  If this is not possible, mark NP and tell why.
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?
8. What whole number less than 50 has the largest number of factors?
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?
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.)
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?
12. A pizza company advertises that its 18-inch (diameter) party pizza has more pizza that two medium (12-inch) pizzas.  Are they right?
13. 
14. 
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?
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?
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.
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?

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 ?

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.)

Bonus #3  See #8 above.  Can you find the smallest whole number with exactly 10 factors?

Bonus #4:  See # 11 above.  What if the original square has side x ?

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