Euler found four whole numbers such that the sum of any two of the
numbers is a perfect square. Three of the numbers that he found are
a = 18530, b=65570, c=45986. Find the fourth number, x. You could
do this by trial and error, and a spreadsheet would be a good tool
for such work. Write down a+x = P^2, b+x = Q^2, c+x = R^2, and then
focus on Q^2-R^2=b-c which is known. Moreover you know that Q >
sqrtb and R > sqrtc . Use this to show that Q-R is less than or
equal to 41 . Use a spreadsheet to calculate values of Q+R , Q and
x for values of Q-R from 1 to 41 , and hence to find the value of x
for which a+x is a perfect square.
The diagram illustrates the formula: 1 + 3 + 5 + ... + (2n - 1) = n² Use the diagram to show that any odd number is the difference of two squares.
Find the frequency distribution for ordinary English, and use it to help you crack the code.
First the fire :
If, just as a first approximation, we don't worry about a filled
bucket being heavier to carry, what is the best point on the river
bank for the fire-fighter to fill the bucket ?.
If you need to do a calculation with lengths, what measurements
will you need to make from your diagram ?
It's a general solution you are looking for, so you may need two
or three different arrangements or diagrams to see how the solution
relates to the positions of the fire-fighter and the fire.
Now the keys :
Draw a horizontal line. Fix two pins at different horizontal
levels above the line. A set of keys slides on a string and the
string runs over those two pins (the pins are not directly
underneath one another). Gradually let out the string length until
the weight of the keys brings the string down to touch the drawn
Can you see the connection between this problem and the fire
problem above ?