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.
Invite ideas about possible directions for generalisation,
perhaps starting with the easier results like allowing 'plus one
more' to become plus two, plus three, and so on.
More able students will produce more extended generalisations
and have a motivation to account for what is observed, challenging
one another to communicate clear explanations or visualisations of
the fundamental processes.
Able students will sense the potential power of a spreadsheet
and should be encouraged to work collaboratively to become
proficient and confident in its use.