May/June ’18 Brain Teasers – SUMMARY
REMINDER: Answers in red. Solvers (submitted/correct) in blue. (Forgive any omissions.) Comments in green. For further elaboration, feel free to ask!
1. A given cube has the length of each side doubled to form a new cube. How does the volume of the new cube relate to the old one? The volume of the new cube is 8 times that of the old one. Jim Waterman
2. In BT #1 above, how is the surface area of the new cube changed? The surface area of the new cube is 6 times that of the old one Jim Waterman
3. 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. Jim Waterman
4. See #3. 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 Jim Waterman
5. Which, if any, of these numbers is the greatest? π, 3.14, 22/7 22/7 is largest. Jim Waterman
6. A) When, if ever, will a positive number be less than its square? When it is less than 1. B) When, if ever, will the cube of any number be less than its square? When it is less than 1 and NOT between 0 and -1. (A tiny bit tricky. Think about it.) C) When, if ever, will a number, its square, and its cube all be equal? When the number is either 0 or 1. Ji.m Waterman (partial credit).
7. Connie Cointosser shakes up a dollar’s worth of quarters and lets them fall on the floor. How likely is it that she gets two heads and two tails? It will happen 6/16 = 3/8 = 37.5% of the time. (There are 16 possible outcomes for four coins. Only 6 are two H, two T.) Jim Waterman
8. Pick any college or university in the United States. How large does the enrollment have to be to guarantee that at least two people on campus somewhere that have the same first and last initials for their names? 677. (26 x 26 = 676, so need 677 to guarantee duplicate.) Jim Waterman (partial credit)
None of #9 – #12 had solutions submitted (a first!!). So, we’ll hold #9 over for one more round (see Jul/Aug BTs), and punt on the rest of them. 🙁
9. GEOGRAPHY FUN: A) Missouri is tied with one other state for the title of ‘state with borders the most other states.’ Can you name the other state, and can you tell how many states they border? B) Missouri is also one of EIGHT states that start with the letter M. Can you name the other seven? C) There are also eight states beginning with the letter N, but they kinda have to ‘cheat’ to do it? Why is that? See July/Aug BTs.
10. A traditional die is rolled. How likely is it that the number shown will be prime? 2, 3, and 5 are prime, so 3/6 or 1/2. How likely that it will be either a factor or a multiple of 3? 1, 3, and 6 meet the conditions, so again 3/6 or 1/2.
11. Two traditional dice are rolled. How likely is it that both numbers showing are prime? 9/36 or 1/4. How likely that at least one is? 27/36 = 3/4. (This one’s trickier. There are only 9 (of 36) outcomes in which NEITHER is prime, so all others (that is the 27 others) have at least one prime.)
12. A traditional die is rolled and one card is drawn from a ‘regular’ 52-card deck. (No jokers). How likely is it that both dice and the drawn card are all 6’s? 1/468. (1/6 x 1/6 x 1/13)
BONUSES:
B1. If a number x is between 3 & 4, find another number y, also between 3 & 4, that is less than x. (Be careful. Perhaps a hint next time.) For example, (3 + x)/2. Jim Waterman (partial credit).
No solutions submitted for B2 – B4. They will perhaps be recycled. If questions, feel free to ask.
B2. Take any two fractions a/b and c/d. (Let’s keep them both positive for now.) Now add the numerators and denominators and make a new fraction (a+c)/(b+d). How often will that new fraction be between the original two fractions? (Example: Suppose the fractions are 1/2 and 1/4. Note that, in this case, (1+1)/(2+4) = 2/6 = 1/3 is between the starting two fractions.)
B3. See #8 above. How large does the enrollment have to be to guarantee the same thing, also using middle initial?
B4. See Question #11. Answer either question there for THREE traditional dice.
