Puzzles
1089
Take a three digit number. Reverse the digits then take the smaller number from the larger number.
Next add the answer to its reverse.
For example, if 175 is chosen:
$$571-175=396$$
$$396+693=1089$$
What numbers is it possible to obtain as an answer, and when will each be obtained?
Dirty work
Source: Futilty Closet
Timothy, Urban, and Vincent are digging identical holes in a field.
When Timothy and Urban work together, they dig 1 hole in 4 days.
When Timothy and Vincent work together, they dig 1 hole in 3 days.
When Urban and Vincent work together, they dig 1 hole in 2 days.
Working alone, how long does it take Timothy to dig one hole?
3n+1
Let \(S=\{3n+1:n\in\mathbb{N}\}\) be the set of numbers one more than a multiple of three.
(i) Show that \(S\) is closed under multiplication.
ie. Show that if \(a,b\in S\) then \(a\times b\in S\).
Let \(p\in S\) be irreducible if \(p\not=1\) and the only factors of \(p\) in \(S\) are \(1\) and \(p\). (This is equivalent to the most commonly given definition of prime.)
(ii) Can each number in \(S\) be uniquely factorised into irreducibles?
2009
Source: Teaching Further Maths blog
2009 unit cubes are glued together to form a cuboid. A pack, containing 2009 stickers, is opened, and there are enough stickers to place 1 sticker on each exposed face of each unit cube.
How many stickers from the pack are left?
Triangles between squares
Prove that there are never more than two triangle numbers between two consecutive square numbers.
Twenty-one
Scott and Virgil are playing a game. In the game the first player says 1, 2 or 3, then the next player can add 1, 2 or 3 to the number and so on. The player who is forced to say 21 or above loses. The first game went like so:
Scott: 3
Virgil: 4
Scott: 5
Virgil: 6
Scott: 9
Virgil: 12
Scott: 15
Virgil 17
Scott: 20
Virgil: 21
Virgil loses.
To give him a better chance of winning, Scott lets Virgil choose whether to go first or second in the next game. What should Virgil do?
Polya strikes out
Write the numbers 1, 2, 3, ... in a row. Strike out every third number beginning with the third. Write down the cumulative sums of what remains:
1, 2, 3, 4, 5, 6, 7, ...
1, 2, 3, 4, 5, 6, 7, ...
1, 2, 4, 5, 7, ...
1=1; 1+2=3; 1+2+4=7; 1+2+4+5=12; 1+2+4+5+7=19; ...
1, 3, 7, 12, 19, ...
Now strike out every second number beginning with the second. Write down the cumulative sums of what remains. What is the final sequence? Why do you get this sequence?
Exact change
Source: @AlexDBolton on Twitter
In the UK, the coins less than £1 are 1p, 2p, 5p, 10p, 20p and 50p. How many coins would I need to carry in my pocket so that I could make any value from 1p to 99p?
In the US, the coins less than $1 are 1¢, 5¢, 10¢, 25¢. How many coins would I need to carry in my pocket so that I could make any value from 1¢ to 99¢?