Peculiar Sequence of Words

What is the peculiar property of this sequence of words?


If you take the first letter from each word, you will spell the first word in the sequence.
If you take the second letter from each word, you will spell the second word in the sequence.
If you take the third letter from each word, you will spell the third word in the sequence.
If you take the fourth letter from each word, you will spell the fourth word in the sequence.


Dead Man in a Hut

One day, the police found a man dead inside a hut. In his left hand, he was holding a gun. In his right hand, he had a recording device. When the recording was played, the police heard the man talking about how horrible his life was and how he wanted it to end. The recording ended with a gunshot. The police was about to call it a suicide until one detective pointed out a very important clue. What was it?

If the man shot himself with the gun, then he would not have been able to stop recording with the device.

Splitting Coins

You split 1000 coins into two piles and count the number of coins in each pile. If there are X coins in pile one and Y coins in pile two, you multiple the two numbers to get XY. Then you split both piles further, repeating the same counting and multiplication process, and adding the new multiplication results to the first one. The process ends when you end up with 1000 single-coin piles. Prove that you will always get the same final result, no matter how the piles have been divided during the splitting process.

For example, if you start with 5 coins and split them into a pile of 2 and a pile of 3, you get the number 2×3=6. Then, if you split the pile of 3 into a pile of 1 and a pile of 2, you will add 1×2=2 more to the 6 and get 8. Finally, if you split the two piles of 2 into single-coin piles, you will end up with 8+1+1=10.

Consider the sum of the squares of the numbers of coins in each pile, plus twice the sum of the products. On each step, if you split a pile of X+Y coins into a pile of X coins and a pile of Y coins, the sum of the squares will get reduced by 2XY, exactly the amount the sum of the products will increase by. Therefore, that number remains constant throughout the entire process and ends up exactly (1000²-1000)/2=499500.

Sequence 1, 3, 7, 12

Which is the next number in the following sequence:

1, 3, 7, 12, 18, 26, 35, 45, 56, ?

This is the so called Hofstadter Figure-Figure Sequence.

The sequence of the differences between the consecutive numbers in the original sequence is 2, 4, 5, 6, 8, 9, 10, 11… These are exactly the natural numbers missing from the original sequence. Therefore, the next number should be 56 + 13 = 69.