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.
Many people thought that the middle
position would be best - that the cube size which only just fits
under in the middle would not fit under anywhere else . . . .
however no one has yet offered a reason for believing this . .A
spreadsheet may help, though only if we can reason correctly about
what the spreadsheet shows. You may like to look at a spreadsheet
created for this problem.
Excel file : UnderRibbon
First you need to select a size of cube, the
final column will then show the amount of ribbon needed for that
cube placed at various positions along the metre length. However
you will still need to reason quite carefully before you can be
sure that the centre is the best position.Once you are happy about
that, the Excel file has a second sheet called 'Cube in the Middle'
(see the blue tab at the bottom of the page).This sheet represents
a 'trial and improvement' method for a cube placed at the
mid-point. You select the cube size to start at and also the size
of the increase row on row, the final column will report the total
length of ribbon required. That needs to be as close to 104 as you
can manage without exceeding it. Although no one really accounted
for their choice of the middle position, some good work was done
with that position assumed.