Puzzles
Fair dice
Source: Futility Closet
Timothy and Urban are playing a game with two six-sided dice. The dice are unusual: Rather than bearing a number, each face is painted either red or blue.
The two take turns throwing the dice. Timothy wins if the two top faces are the same color, and Urban wins if they're different. Their chances of winning are equal.
The first die has 5 red faces and 1 blue face. What are the colours on the second die?
Half digits
Source: Maths Jam
Can you use each of the digits 1 to 9 to make a fraction which is equal to a half?
Pizza
Twelve friends want to share a pizza. One of the friends is very fussy and will not eat the centre of the pizza.
Is it possible to split a (circular) pizza into twelve identical pieces such that there is at least one piece which does not touch the centre?
![](javascript:showlimage("pizza-answer.png"))
Frogs
Source: nrich
Two frogs and two toads are standing on five lily pads.
![](javascript:showlimage("frogs-and-toads.png"))
The frogs and toads need to pass each other. They can only move by jumping one or two lily pads forward. In jumping two pads forwards they can jump over other frogs or toads.
How many jumps need to be made to get the frogs and toads past each other?
The blue-eyed sisters
If you happen to meet two of the Jones sister (two sisters chosen at random from all the Jones sisters), it is exactly an even-money bet that both will be blue-eyed. What is your best guess of the total number of Jones sisters?
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?
Integrals
$$\int_0^1 1 dx = 1$$Find \(a_1\) such that:
$$\int_0^{a_1} x dx = 1$$
Find \(a_2\) such that:
$$\int_0^{a_2} x^2 dx = 1$$
Find \(a_n\) such that (for \(n>0\)):
$$\int_0^{a_n} x^n dx = 1$$