Jose, Thuy, and Kareem each start with the number 10. Jose subtracts 1 from 10, doubles his answer, and then adds 2. Thuy doubles 10, subtracts 1 from her answer, and then adds 2. Kareem subtracts 1 from 10, adds 2 to his number, and then doubles the result. Who gets the largest final answer?
The 64 whole numbers from 1 through 64 are written, one per square, on a checkerboard (an 8 by 8 array). The first 8 numbers go in order across the first row, the next 8 across the second row, and so on. After all 64 numbers are written, the sum of the numbers in the four corners will be
Show hint (soft nudge)
The top row is 1–8 and the bottom row is 57–64.
Show hint (sharpest)
The corners are the first and last entry of those two rows.
Show solution
The corners are 1 and 8 (top row) and 57 and 64 (bottom row).
What is the smallest result that can be obtained from the following process? Choose three different numbers from the set {3, 5, 7, 11, 13, 17}, add two of them, then multiply their sum by the third number.
Show hint (soft nudge)
To make a product small, multiply by the smallest number.
Show hint (sharpest)
Then add the two next-smallest numbers.
Show solution
Use 3 as the multiplier and add the next two smallest, 5 and 7: (5 + 7) × 3 = 36.
Any other choice (such as (3 + 7) × 5 = 50) is larger, so the smallest is 36.
Brent has goldfish that quadruple (become four times as many) every month, and Gretel has goldfish that double every month. If Brent has 4 goldfish at the same time that Gretel has 128 goldfish, in how many months from that time will they have the same number of goldfish?
Show hint (soft nudge)
Write both counts as powers of 2 and match the exponents.
Show hint (sharpest)
Brent: 4 · 4^m; Gretel: 128 · 2^m.
Show solution
After m months Brent has 4 · 4^m = 2^(2m+2) and Gretel has 128 · 2^m = 2^(m+7).
Points A and B are 10 units apart. Points B and C are 4 units apart. Points C and D are 3 units apart. If A and D are as close as possible, then the number of units between them is
Show hint (soft nudge)
Walk from A: out 10, then come back as much as possible.
Show hint (sharpest)
The best you can backtrack is 4 + 3 = 7.
Show solution
Place B 10 from A, then step C back 4 and D back another 3: A is at 0, D ends at 10 − 4 − 3 = 3.
You can't reach 0 (10 ≠ 4 + 3), so the closest is 3 units.
When Walter drove up to the gasoline pump, his tank was 1/8 full. He bought 7.5 gallons, after which the tank was 5/8 full. How many gallons does the tank hold when it is full?
Show hint (soft nudge)
The 7.5 gallons raised the level from 1/8 to 5/8 — what fraction of the tank is that?
Show hint (sharpest)
Then scale up to a full tank.
Show solution
Going from 1/8 to 5/8 is 4/8 = half the tank, filled by 7.5 gallons.
Let x be the number 0.00…01, where there are 1996 zeros after the decimal point before the 1. Which of the following expressions represents the largest number?
Show hint (soft nudge)
x is an extremely tiny positive number.
Show hint (sharpest)
Dividing by something tiny makes the result enormous.
Show solution
Since x is a tiny positive number, 3 + x, 3 − x, and 3·x are all near 3 (or near 0), and x/3 is tiny.
But 3/x divides by something tiny, giving a gigantic number, so 3/x is largest.
In the fall of 1996, 800 students took part in an annual school clean-up day. The organizers expect that in each of 1997, 1998, and 1999, participation will increase by 50% over the previous year. The number of participants expected in the fall of 1999 is
Show hint (soft nudge)
A 50% increase multiplies by 1.5 each year.
Show hint (sharpest)
Apply that three times to reach 1999.
Show solution
Three yearly increases multiply by 1.5 three times: 800 × (3/2)³ = 800 × 27/8.
Ana's monthly salary was $2000 in May. In June she received a 20% raise. In July she received a 20% pay cut. After the two changes in June and July, Ana's monthly salary was
Show hint (soft nudge)
A 20% raise then a 20% cut do NOT cancel out.
Show hint (sharpest)
Multiply by 1.2, then by 0.8.
Show solution
After the raise: 2000 × 1.2 = 2400. After the cut: 2400 × 0.8 = 1920.
A special key on a calculator replaces the displayed number x with 1 ÷ (1 − x). (For example, from 2 it gives 1 ÷ (1 − 2) = −1.) If the calculator shows 5 and the key is pressed 100 times in a row, the calculator will display
Show hint (soft nudge)
Press the key a few times and watch for a repeating cycle.
Show hint (sharpest)
Once you know the cycle length, reduce 100 by it.
Show solution
Starting at 5: 5 → −0.25 → 0.8 → 5, a cycle of length 3.
Since 100 = 3·33 + 1, after 100 presses the display matches one press: −0.25.
How many subsets containing three different numbers can be selected from the set {89, 95, 99, 132, 166, 173} so that the sum of the three numbers is even?
Show hint (soft nudge)
Only the parity (odd/even) of each number matters: there are 4 odds and 2 evens.
Show hint (sharpest)
A sum of three is even only when an even count of them are odd — here that means exactly 2 odds and 1 even.
Show solution
The set has 4 odd numbers (89, 95, 99, 173) and 2 even (132, 166). Three numbers sum to even only with 2 odds and 1 even (zero odds would need 3 evens, impossible).
The manager of a company planned to give a $50 bonus to each employee from the company fund, but the fund was $5 short of what was needed. Instead the manager gave each employee a $45 bonus and kept the remaining $95 in the fund. How much money was in the company fund before any bonuses were paid?
Show hint (soft nudge)
Let n be the number of employees and write the fund two ways.
Show hint (sharpest)
Fund = 50n − 5 (just short of $50 each) and fund = 45n + 95 (after the $45 bonuses).
Show solution
The fund is 50n − 5 (5 short of $50 each) and also 45n + 95 (gave $45 each, kept $95).
Setting them equal: 50n − 5 = 45n + 95 gives n = 20, so the fund was 45·20 + 95 = $995.
A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region than it is to the boundary of the region?
Show hint (soft nudge)
A point at distance r from the center is r from the center and R − r from the boundary.
Show hint (sharpest)
Closer to the center means r < R − r, i.e. r < R/2.
Show solution
Being closer to the center than the boundary means r < R − r, so r < R/2 — the point lies in the inner circle of radius R/2.
Its area is a fraction (R/2)² / R² = 1/4 of the whole.
