Puzzles
Turning squares
Source: Futility Closet
Each square on a chessboard contains an arrow point up, down, left or right. You start in the bottom left square. Every second you move one square in the direction shown by the arrow in your square. Just after you move, the arrow on the square you moved from rotates 90° clockwise. If an arrow would take you off the edge of the board, you stay in that square (the arrow will still rotate).
You win the game if you reach the top right square of the chessboard. Can I design a starting arrangement of arrows that will prevent you from winning?
Elastic numbers
Throughout this puzzle, expressions like \(AB\) will represent the digits of a number, not \(A\) multiplied by \(B\).
A two-digit number \(AB\) is called elastic if:
- \(A\) and \(B\) are both non-zero.
- The numbers \(A0B\), \(A00B\), \(A000B\), ... are all divisible by \(AB\).
There are three elastic numbers. Can you find them?
Square pairs
Source: Maths Jam
Can you order the integers 1 to 16 so that every pair of adjacent numbers adds to a square number?
For which other numbers \(n\) is it possible to order the integers 1 to \(n\) in such a way?
Factorial pattern
$$1\times1!=2!-1$$ $$1\times1!+2\times2!=3!-1$$ $$1\times1!+2\times2!+3\times3!=4!-1$$Does this pattern continue?
Placing plates
Two players take turns placing identical plates on a square table. The player who is first to be unable to place a plate loses. Which player wins?
Advent 2016 murder mystery
2016's Advent calendar ended with a murder mystery, with each of
the murderer, motive, weapon and location being a digit from 1 to 9.
Here are the clues:
10
None of the digits of 171 is the location.
None of the digits of 171 is the location.
3
None of the digits of 798 is the motive.
None of the digits of 798 is the motive.
7
One of the digits of 691 is the location.
One of the digits of 691 is the location.
16
None of the digits of 543 is the location.
None of the digits of 543 is the location.
5
One of the digits of 414 is the murderer.
One of the digits of 414 is the murderer.
20
The first digit of 287 is the number of false red clues.
The first digit of 287 is the number of false red clues.
8
Clues on days that are factors of 768 are all true.
Clues on days that are factors of 768 are all true.
22
The murderer is the square root of one of the digits of 191.
The murderer is the square root of one of the digits of 191.
11
One of the digits of 811 is the weapon.
One of the digits of 811 is the weapon.
19
The highest common factor of the weapon and 128 is 1.
The highest common factor of the weapon and 128 is 1.
13
None of the digits of 512 is the murderer.
None of the digits of 512 is the murderer.
18
One of the digits of 799 is the motive.
One of the digits of 799 is the motive.
17
None of the digits of 179 is the motive.
None of the digits of 179 is the motive.
6
None of the digits of 819 is the location.
None of the digits of 819 is the location.
24
One of the digits of 319 is total number of false clues.
One of the digits of 319 is total number of false clues.
23
One of the digits of 771 is the murderer.
One of the digits of 771 is the murderer.
2
The weapon is not one of the digits of 435.
The weapon is not one of the digits of 435.
14
The final digit of 415 is the number of true blue clues.
The final digit of 415 is the number of true blue clues.
4
The weapon is a factor of 140.
The weapon is a factor of 140.
12
The number of false clues before today is the first digit of 419.
The number of false clues before today is the first digit of 419.
9
One of the digits of 447 is the motive.
One of the digits of 447 is the motive.
1
None of the digits of 563 is the motive.
None of the digits of 563 is the motive.
21
One of the digits of 816 is the murderer.
One of the digits of 816 is the murderer.
15
One of the digits of 387 is the motive.
One of the digits of 387 is the motive.
24 December
Today's number is 191 more than one of the other answers and
100 less than another of the answers.
23 December
Today's number is the number of three digit numbers that are not three more than a multiple of 7.