Group the 27 small cubes by where they sit: 8 corners, 12 edges, 6 face-centers, 1 internal.
Show hint (sharpest)
On each face, only the very center square sits at a face-center sub-cube — and only those (plus the internal cube) escape being shaded.
Show solution
Each face is shaded everywhere except its center square. So 8 corner sub-cubes and 12 edge sub-cubes each show at least one shaded square; only the 6 face-center sub-cubes and the 1 internal cube show none.
8 + 12 = 20 small cubes have at least one shaded face.
Add the whole parts first, then bound the fractions.
Show hint (sharpest)
1⁄7 + 1⁄2 + 1⁄19 — at most 3⁄2, at least 1⁄2.
Show solution
Whole parts: 2 + 3 + 5 = 10. Fractions: 1⁄7 + 1⁄2 + 1⁄19 sits between 1⁄2 and 1 (clearly more than 1⁄2 since one term is already 1⁄2, but less than 1 since 1⁄7 + 1⁄19 < 1⁄2).
Total between 10 + 1⁄2 = 10 1⁄2 and 10 + 1 = 11. Answer B.
The sale ad read: "Buy three tires at the regular price and get the fourth tire for three dollars." Sam paid 240 dollars for a set of four tires at the sale. What was the regular price of one tire?
Show hint
Subtract the $3 promo tire first, then split what's left across three tires.
Show solution
Three full-price tires cost 240 − 3 = 237 dollars.
Joyce made 12 of her first 30 shots in the first three games of this basketball game, so her seasonal shooting average was 40%. In her next game, she took 10 shots and raised her seasonal shooting average to 50%. How many of these 10 shots did she make?
Show hint (soft nudge)
After the next game her total shots is 40 and average is 50%, so total made = 20.
Show hint (sharpest)
She'd already made 12 — how many more did she need?
Show solution
After 40 shots at 50%, she's made 20. She had 12 before, so she made 20 − 12.
Abby, Bret, Carl, and Dana are seated in a row of four seats numbered #1 to #4. Joe looks at them and says:
"Bret is next to Carl."
"Abby is between Bret and Carl."
However each one of Joe's statements is false. Bret is actually sitting in seat #3. Who is sitting in seat #2?
Show hint (soft nudge)
Both statements are false — turn each one into a 'not'.
Show hint (sharpest)
If Bret (#3) and Carl aren't adjacent, Carl must be in seat #1.
Show solution
Bret is in #3, and "Bret next to Carl" is false, so Carl isn't in #2 or #4 — Carl is in #1. Then "Abby between Bret and Carl" being false rules Abby out of #2, the only spot strictly between them.
Half the people in a room left. One third of those remaining started to dance. There were then 12 people who were not dancing. The original number of people in the room was what?
Show hint (soft nudge)
After the leavers, half remain; of those, 2⁄3 are not dancing.
Show hint (sharpest)
(1⁄2) × (2⁄3) × N = 12.
Show solution
Non-dancers are 1⁄2 × 2⁄3 = 1⁄3 of the original group.
A multiple choice examination consists of 20 questions. The scoring is +5 for each correct answer, −2 for each incorrect answer, and 0 for each unanswered question. John's score on the examination is 48. What is the maximum number of questions he could have answered correctly?
Show hint (soft nudge)
Let c = correct, w = wrong; 5c − 2w = 48 with c + w ≤ 20 and c, w ≥ 0 integers.
Show hint (sharpest)
5c must be even, so c must be even — that limits c to even values.
Show solution
From 5c − 2w = 48, c must be even (so 5c is even); from c + w ≤ 20, c is at most about 12.6.
The biggest even c that fits is c = 12 (w = 6, total 18).
Ten balls numbered 1 to 10 are in a jar. Jack reaches into the jar and randomly removes one of the balls. Then Jill reaches into the jar and randomly removes a different ball. The probability that the sum of the two numbers on the balls removed is even is
Show hint (soft nudge)
The sum is even when both balls have the same parity.
Show hint (sharpest)
5 odd and 5 even balls; second draw is without replacement, so the denominator becomes 9.
