35 Moves

Design a game that takes less than 35 moves to get to the position below.

One possible solution is:

1.d3 h6 2.Bxh6 f5 3.Qd2 f4 4.Qxf4 a5 5.Qxc7 Kf7 6.g3 Kg6 7.Bg2 Kh5 8.Bxb7 Kg4 9.Nf3 Kh3 10.Bxc8 e5 11.Bxg7 e4 12.Kd2 e3+ 13.Kxe3 Kg2 14.Ng1 Kf1 15.Kf3 Ke1 16.Qxa5+ Bb4 17.Nc3+ Kd2 18.Rf1 Rh3 19.Bxd7 Nh6 20.Nd1 Kc1 21.Bxh6+ Kb1 22.Bc1 Na6 23.Kg2 Rc8 24.Bxh3 Rc3 25.Nxc3+ Ka1 26.Nb1 Nc5 27.Rd1 Be1 28.Qxe1 Ne4 29.Kf1 Nd2+ 30.Rxd2 Qd5 31.Qd1 Qg2+ 32.Ke1 Qf1+ 33.Bxf1

Vowels and Even Numbers

“If there is a vowel written on one side of a card, then there is an even number written on the other side.”
How many of these four cards do you need to flip in order to check the validity of this sentence?

What would the answer be if you know that each card contains one letter and one number?

You need to flip all cards except for the second one. If each card contains one letter and one number, then you need to flip only A and 7.

Progression Banned

I give you a pen and paper and ask you to write the numbers from 1 to 100 in succession so that there are no three numbers such that twice the second one is equal to the sum of the first and the third one. The three numbers do not need to be successive in the sequence.

You have 5 minutes, what do you do?

Remark: The sequence 3, 1, 2, 5, 4 works, but the sequence 1, 4, 2, 5, 3 does not because of the numbers 1, 2, and 3.

Start with the following sequences:

1  →  1, 2  →  2, 4, 1, 3  →  4, 8, 2, 6, 3, 7, 1, 5  →  8, 16, 4, 12, 6, 14, 2, 10, 7, 15, 3, 11, 5, 13, 1, 9

and keep iterating until you get a sequence with all numbers from 1 to 128. On each step you take the previous sequence, multiply all elements by 2, and then add the same result but with all elements decreased by 1. This will ensure that the first half contains only even numbers and the second half contains only odd numbers. Since the sum of an odd and an even number is not divisible by 2, if some sequence violates the property, then the previous sequence would have violated it as well.

Once you construct a sequence with 128 numbers, simply remove the numbers from 101 to 128 and you are done. To speed up the process, you can reduce the sequence 8, 16, 4, 12, 6, 14, 2, 10, 7, 15, 3, 11, 5, 13, 1, 9 to 8, 4, 12, 6, 2, 10, 7, 3, 11, 5, 13, 1, 9 and then continue the process.

Special Transaction

One person went to the store and bought groceries for $13.59 total. He paid with a $100 bill, took his change, and left the store. There was something special about this transaction. What is it?

The person paid with a $100 bill. The cashier returned him a $50 bill, a $20 bill, a $10 bill, a $5 bill, a $1 bill, a quarter, a dime, a nickel, and a cent. The transaction consisted of exactly one of each (frequently used) denominations.

1985

What number corresponds to 1985?

0 0 0 0 – 4
1 7 5 2 – 0
1 8 7 9 – 3
2 0 6 1 – 2
3 1 4 1 – 0
4 0 9 6 – 3
7 7 7 7 – 0
9 9 7 3 – 2
1 9 8 5 – ???

The numbers on the right count the total number of “holes” in the digits on the left. “1”, “2”, “3”, “4”, “5” and “7” have 0 holes in them. “0”, “6” and “9” have 1 hole in them. “8” has 2 holes in it. Therefore, the corresponding number is 3.