Random Puzzles 

Workings

Hint
How many miles were left at the start of yesterday?

Answer
336 miles.

Reasoning
To work these numbers out, it is perhaps easier to start at the end and work backwards.

Yesterday I travelled ¹/₃ of the distance to leave 21 miles, therefore there were 31.5 miles (=21 ÷ ²/₃) at the start of yesterday.

On day 3 I travelled ³/₄ of the distance to leave 31.5 miles, therefore there were 126 miles (=31.5 ÷ ¹/₄) at the start of day 3.

On day 2 I travelled ¹/₂ of the distance to leave 126 miles, therefore there were 252 miles (=126 ÷ ¹/₂) at the start of day 2.

On day 1, I travelled ¹/₄ of the distance to leave 252 miles, therefore there were 336 miles (=252 ÷ ³/₄) at the start of day 1.

Double-Checking
On day one, I travelled one quarter of the distance (84, leaving 252).

On day two, I travelled one half of the remaining distance (126, leaving 126).

On day three, I travelled three quarters of the remaining distance (94.5, leaving 31.5).

Yesterday I travelled one third of the remaining distance (10.5).

I now have 21 miles left to travel.

Workings

Hint
The first letters of the words are: W, E, S, A, S, C, O, T, W, A.

Answers
Windbreak, eiderdown, sometimes, afternoon, September, carpenter, offensive, telephone, Wednesday, apartment.
win + dbr + eak = windbreak
eid + erd + own = eiderdown
som + eti + mes = sometimes
aft + ern + oon = afternoon
sep + tem + ber = september
car + pen + ter = carpenter
off + ens + ive = offensive
tel + eph + one = telephone
wed + nes + day = wednesday
apa + rtm + ent = apartment

Workings

Hint
Look at Digit 3 first.

Answer
6394.

Reasoning
Labelling the clues:

Digit 2 is smaller than Digit 4 Digit 4 is two thirds of Digit 1 Digit 1 is two thirds of Digit 3 Digit 3 is three times Digit 2
By (3), Digit 3 is divisible by 3:

   Digit 3 = 3, 6, 9

By (3), Digit 1 is two thirds of Digit 3:

   Digit 1 = 2, 4, 6.

But, by (2), Digit 1 is divisible by 3:

   Digit 1 = 6, and Digit 3 = 9.

By (4), Digit 3 is three times Digit 2:

   Digit 2 = 3.

By (2), Digit 4 is two thirds of Digit 1:

   Digit 4 = 4.

Workings

Hint
Don't forget rooms 440-449.

Answer
120 times.

Reasoning
Don't forget rooms 440-449, especially room 444.

400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499.