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?
2. In BT #1 above, how is the surface area of the new cube changed?
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?
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?
5. Which, if any, of these numbers is the greatest? π, 3.14, 22/7
6. A) When, if ever, will a positive number be less than its square? B) When, if ever, will the cube of any number be less than its square? C) When, if ever, will a number, its square, and its cube all be equal?
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?
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?
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?
10. A traditional die is rolled. How likely is it that the number shown will be prime? How likely that it will be either a factor or a multiple of 3?
11. Two traditional dice are rolled. How likely is it that both numbers showing are prime? How likely that at least one is?
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?
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.)
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.
1. 2^3 = 8.
(AXB)^3 = A^3 X B^3
Doubling the side of a cube increases the volume by a multiple of 2^3.
2^3 = 8.
3^3 = 27
(2X3)^3 = 6^3 = 2^3 X 3^3 = 8 X 27 = 216
2. Doubling the side of a cube increases the surface area by a multiple of 2^2 = 4.
Area = 6a^2
A = 6(2a)^2 = 6 X 2^2 X a^2.
A = 6X3^2 = 54.
A = 6 X (2X3)^2 = 6 X 2^2 X 3^2 = 216
3. WBBY
4. RYRG
5. 22/7 = 3.1428571429
Pi = 3.1415926…
3.14 = 3.14
6. A. The number is less than 1. E.G. 1/2 X 1/2 = 1/4
6. B. The number is less than 1. E.G. 1/2 X 1/2 X 1/2 = 1/8
6. C. X = 1. 1 X 1 = 1, 1 X 1 X 1 = 1
B.1. I am tempted to write 3 < 3, but the caution in the question makes me think there is more to this question than meets the eye.
Correction on B.1.
I am tempted to write y < 3, but the question is written in such a way that makes me think there is more to this problem than meets the eye.
Thanks, Jim. READERS: Jim would be right, of course. I neglected to say that y is ALSO between 3 & 4. (I’ve corrected this oversight for future mailings already.)
B.1. This is my first attempt at this question. I reserve the right to change the answer once the hint is delivered.
I am making an assumption about the question, I assume both X & Y are between 3 & 4, but not 3 or 4. That is X & Y are between 3 & 4, not inclusive.
The challenge is to find a value of Y that is less the X, If I start with X = 3.9999, Y is 3 < Y < 3.9999. If X = 3.8, Y is 3 < Y < 3.8. If X = 3.7, Y is 3 < Y < 3.7, etc.
Finally we get to if X = 3.1, 3 < Y < 3.1; X = 3.01, 3 < Y < 3.01; X = 3.001, 3 < Y < 3.001; etc.
As X approaches 3, Y also approaches 3, but neither reaches 3.
This can continue into infinity.
My Answer: If X = 3 + 10 to the power of minus infinity, then 3 < Y < 3 + 10 to the power of minus infinity.
7. 37.5%. There are 16 possible combinations of H/TS for 4 quarters, 6 of them have 2 H and 2 T. 6/16 = 0.375.
8. 651. There are 650 permutations of 2 out of a set of 26.
B.3. 15, 601. There are 15,600 permutations of 3 out of a set of 26.