Happy New Leap Year
2012 is a leap year, i.e. it has 366 days. 366 can be written as the sum of consecutive square numbers:
366 = 82 + 92 + 102 + 112
Non-leap years have 365 days. 365 can be written as the sum of consecutive squares in two different ways:
365 = 102 + 112 + 122
365 = 132 + 142
Neither 366 nor 365 are square numbers themselves. Are there any square numbers that can be written as the sum of consecutive squares?
How odd is Pythagoras?
Pythagorean triples are sets of positive integers (a,b,c) where a2+b2=c2. A Pythagorean triple is primitive if there isn’t a common factor that divides a, b and c.
(3,4,5) and (5,12,13) are primitive Pythagorean triples but (6,8,10) is not as 2 divides 6,8 and 10.
- Is the smallest number in a primitive Pythagorean triple always odd?
- Is the largest number in a primitive Pythagorean triple always odd?
A Shop sells chocolates at £1 each.
You can exchange 3 wrappers for 1 chocolate.
If you have £45, how many chocolates can you get in total?
Take the number 523523.
- Divide it by 13
- Divide your answer by 7
- Divide that answer by 11
- What did you get?
Repeat for 791791.
Can you predict numbers it will work for?
The most important question… why?