Random Puzzles 

Workings

Hint
There are fewer than 1,000 apples.

Answer
946 apples.

Reasoning
We know that (answer − 1) is divisible by 3, 5, 7, and 9.

If it's divisible by 9 then it's automatically divisible by 3.

So the number must be a multiple of 5, 7, and 9.

   [5 x 7 x 9 = 315]

Since these numbers share no other common factors, we know that the answer must be a multiple of 315 (+ 1).

Let's try 1 x 315 + 1: 316 ÷ 11 = 28.73, and this doesn't work.

Let's try 2 x 315 + 1: 631 ÷ 11 = 57.36, and this doesn't work.

Let's try 3 x 315 + 1: 946 ÷ 11 = 86, and this works, and is the answer.

Workings

Hint
Think of the pages from 1 to 9, and then 10 to 99.

Answers
There are 147 pages. The number 3 appears 35 times.
Answer #1
For pages 1 to 9, there are 9 pages, which is 9 digits.

For pages 10 to 99, there are 90 pages, which is 180 digits.

This is a total so far of 189, therefore we require another 333 − 189 = 144 digits, which is another 144 ÷ 3 = 48 pages.

Taking us to 9 + 90 + 48 = 147 pages in total.

Answer #2
From page 1 to 147 we have 15 pages that end in 3:

   3, 13, 23, …, 143

We also have the ten pages that start with 30:

   30, 31, 32, …, 39

Plus the ten pages that start with 130:

   130, 131, 132, …, 139

For a total of 15 + 10 + 10 = 35 number 3's.

Workings

Answer
23.

Reasoning
By likely, we mean greater than 50% chance.

With one person there is a 0 percent chance that you'll have two people with the same birthday.

With two people the probability that they won't share a birthday is 364 ÷ 365. The probability that they will share a birthday is therefore 1 − (364 ÷ 365).

With three people the probability that they won't share a birthday is the same as for two people, times 363 ÷ 365. So the probability that three people will share a birthday is 1 − (364 ÷ 365) x (363 ÷ 365).

Notice that with each additional person added, the probability that they shares a birthday with one of the previous persons goes up, because there are fewer "free" days remaining.

We keep adding people until the %age is greater than 50%.

When we have 23 people the %age is 50.729%.

Workings

Hint
The 8-letter word begins with the letter G.

Answer
Generous.
lend + ge = le + ge + nd = legend

liar + ne = li + ne + ar = linear

pale + ro = pa + ro + le = parole

coin + us = co + us + in = cousin