Hint: Can Q1 be A?

Answer:
1. B
2. B
3. A
4. B

Q1 can't have A as its answer (otherwise it would contradict itself).

If Q1's answer was C (meaning Q4's answer was A), then Q3's answer would be C. However, Q2's answer should now be A, but this isn't allowed by Q1 (as Q4 is the first answer with A). This is a contradiction.

Therefore Q1's answer is B (meaning Q3's answer is A).

For Q2's answer to be C, Q4's answer would have to be A, which would contradict Q4. Therefore Q2's answer is B, making Q4's answer B. QED.

Hint: These are the first letters of the words: T, T, H, D, W, L, H, W, C, N

Answer:

TEL + ESC + OPE = TELESCOPE
TIM + ETA + BLE = TIMETABLE
HAI + RBR + USH = HAIRBRUSH
DIF + FER + ENT = DIFFERENT
WAT + ERF + ALL = WATERFALL
LIG + HTN + ING = LIGHTNING
HEA + DLI + GHT = HEADLIGHT
WHI + RLW + IND = WHIRLWIND
CAR + TIL + AGE = CARTILAGE
NEW + SAG + ENT = NEWSAGENT

Hint: Try breaking each cog size into prime factors.

Answer: 260 revolutions.

There are a number of ways of thinking about the solution, and we find this one the quickest way to find the answer.

The total number of teeth moved by Cog 1 will be wholly divisible by each cog in turn, therefore:

Revolutions x Cog 1 ÷ Cog2 is an integer
Revolutions x Cog 1 ÷ Cog3 is an integer
Revolutions x Cog 1 ÷ Cog4 is an integer

So we are after the first number of revolutions x 81 that is an integer after division by 52, 36 and 20.

Thus:

81     81     81
-- and -- and -- all need to be integers (and not fractions).
52     36     20

An easy way to do this would be to multiply by 52 x 36 x 20 = 37,440 revolutions, which would be a correct answer, but not necessarily the smallest answer.

A better way is to break each cog down into its prime factors, where Cog 1 has the largest number of teeth:

Cog 1 - 81 = 3 x 3 x 3 x 3

Cog 2 - 52 = 2 x 2 x 13
Cog 3 - 36 = 2 x 2 x 3 x 3
Cog 4 - 20 = 2 x 2 x 5

3 x 3 x 3 x 3 and 3 x 3 x 3 x 3 and 3 x 3 x 3 x 3 and these need to be integers
-------------     -------------      ------------
 2 x 2 x 13       2 x 2 x 3 x 3       2 x 2 x 5

Simplifying any fraction that can be simplified gives:

3 x 3 x 3 x 3 and 3 x 3 and 3 x 3 x 3 x 3 and these need to be integers
-------------     -----     -------------
 2 x 2 x 13       2 x 2       2 x 2 x 5

Multiplying throughout by 2 x 2 gives:

3 x 3 x 3 x 3 and 3 x 3 and 3 x 3 x 3 x 3 and these need to be integers
-------------               -------------
     13                            5

Multiplying throughout 13 gives:

3 x 3 x 3 x 3 and 3 x 3 x 13 and 3 x 3 x 3 x 3 x 13 and these need to be integers
                                 ------------------
                                         5

Multiplying throughout 5 gives:

3 x 3 x 3 x 3 x 5 and 3 x 3 x 13 x 5 and 3 x 3 x 3 x 3 x 13

They are all now integers, and we have therefore multiplied by 2 x 2 x 13 x 5.

2 x 2 x 13 x 5 = 260 revolutions. As required.

The easy way from above of 37,440 is exactly 144 times this answer.

Hint: Say what you see.

Answer: One step forward, two steps back.

STEP + PETS + PETS.

Hint: Every word must contain the central letter.

Answer:
Common words: dare, dares, dart, darts, date, dates, day, days, dear, dearest, dears, deer, desert, deter, deters, dray, drays, dry, dye, dyer, dyers, dyes, eared, eased, erased, eyed, rated, read, reads, ready, red, reds, reed, reeds, rested, sad, seared, seated, sedate, seed, seedy, stared, stayed, stead, steady, steed, strayed, tad, tads, tardy, tared, teared, teased, trade, trades, tread, treads, yard, yards, yeasted, YESTERDAY.

All words: ad, ads, aedes, dare, dares, dart, darts, date, dater, daters, dates, day, days, de, dear, dearest, dears, deary, dee, deer, deers, dees, deet, deets, derat, derate, derates, derats, deray, derays, dere, desert, deter, deters, dey, deys, dray, drays, dree, drees, drest, dry, dryest, drys, dye, dyer, dyers, dyes, eared, eased, ed, erased, estrayed, eyed, rad, rads, rased, rated, rayed, read, reads, ready, red, redate, redates, rede, redes, reds, redye, redyes, reed, reeds, reedy, reseda, rested, sad, sade, sard, sated, seared, seated, sedate, sedater, seder, seed, seedy, sered, stade, stared, stayed, stead, steady, steed, strayed, styed, tad, tads, tardy, tared, teared, teased, ted, teds, teed, trad, trade, trades, tread, treads, treed, tsade, tyred, yard, yards, yeasted, YESTERDAY.