4. Initially there are m balls in one bag, and n in the other, where m, n >
advised to concentrate on these problems first.
0. Two different operations are allowed:
• The use of rulers and compasses is allowed, but
a) Remove an equal number of balls from each bag;
calculators and protractors are forbidden.
b) Double the number of balls in one bag.
• Start each question on a fresh sheet of paper. Write
Is it always possible to empty both bags after a finite sequence of
on one side of the paper only. On each sheet of
operations?
working write the number of the question in the
top left hand corner and your name, initials and
Operation b) is now replaced with
school in the top right hand corner.
b
′) Triple the number of balls in one bag.
• Complete the cover sheet provided and attach it to
Is it now always possible to empty both bags after a finite sequence of
the front of your script, followed by your solutions
in question number order.
• Staple all the pages neatly together in the top left
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