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.
One reason for using this problem with students is to show how a
spreadsheet approach can give access to problems where other
analysis, in this case algebra, is very difficult.
A second reason is that the answer to the problem is
Somehow the brain doesn't expect such a large answer - perhaps
we imagine that because there isn't much extra length extending the
ribbon ( 4 cm of slack ) there will not be much scope for
displacing the ribbon vertically.
This problem also needs approaching in stages, and it is good
for students to experience that by solving for the symmetric
central position first then giving consideration to the asymmetric
Perhaps students could
extend the enquiry to consider how different degrees of
slack : 3 cm, 2 cm, 1 cm, effect the result.
Some students will be very familiar with using a spreadsheet to
help with this kind of mathematical context, but many will not be
at all confident, or even think to try a spreadsheet.
Making calculations for several different cube sizes on a
calculator first, helps students to see that the calculation
process is the same each time only using different input values. It
also helps students to grasp that the ability to cover input values
at intervals systematically across a chosen range could guide us
closer and closer towards an optimal position or value.