Random Puzzles 

Workings

Hint
There are two different methods, one involves a cube root, and the other doesn't require a calculator.

Answer
9 x 11 x 13.

Reasoning #1
We're after three numbers that multiply together, so a good place to start is the cube root of 1,287, which is roughly 10.88.

Let's try dividing by the closest odd number to 10.88:

   1287 ÷ 11 = 117

We're now after whole divisors of 117. Trying the odd numbers either side of 11 might work.

Trying either 9 or 13 gives the answer:

   9 x 11 x 13 = 1,287

Trying the closest odd number to the cube root always works, and the other two numbers are the odd numbers either side.

Reasoning #2
We after three numbers that multiply together, but none of these can end in 5 (otherwise our answer would end in 0 or 5).

So, they can only end in 7, 9, 1 (e.g. 87, 89, 91), or 9, 1, 3 (e.g. 89, 91, 93).

However, if they ended in 7, 9, 1, the answer would end in 3 (because 7 x 9 x 1 = 63).

Therefore, they end in 9, 1, and 3 (because 9 x 1 x 3 = 27).

The first numbers we can try are 9, 11, 13:

   = 9 x 11 x 13
   = 99 x 13
   = 100 x 13 - 13
   = 1300 - 13
   = 1,287

A calculator is not required!

Workings

Hint
Pages 1-9 are one digit each.

Answer
There are 65 pages and the number 9 appears six times.

Reasoning
The 65 pages are (with the six 9's highlighted):

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65.

If we remove the spaces, we get:

1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950515253545556575859606162636465. Which is 121 digits.

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.

Workings

Hint
Loot at each number backwards.

Answer
32.

The first prime numbers reversed.