Here's a spreadsheet file that helps - Peaches Generally
There are three pages or sheets in this file (look for the coloured
tabs at the bottom)
The first called 'Forwards' shows the basic Peaches problem and the
solution of 22 is picked out in yellow.
The second sheet is called 'Backwards' which starts with the
end number of peaches and works back to the start number.
This is more efficient. Can you see why ?
The third sheet is called 'with Variables' and will help you
The general process is to have repetitions of two alternating
actions. One action is to multiply by a constant amount. In the
basic Peaches problem, that multiplier is 0.5, when half the
peaches remain. The second action is to add, or subtract, a
constant amount. That value is -1 in the original setup.
Experiment varying these two values. You should quickly find some
patterns to think about.
What's going on, and why does it happen ?
You may also notice that there isn't really a 'forwards' or
'backwards', just a line of values and a repeated pair of actions
which takes you along the line in the direction you choose.
Choosing the opposite direction just uses the inverses of those
actions and in the opposite order.
Once you have a really good feel for this you might like to
find out about Repayment Mortgages.
There's a multiplier, involving the interest rate, and a constant
addition, the amount of the loan paid off each month.
That addition is negative because the loan amount, which is the
quantity we really care about, is reduced rather tha increased by
each monthly payment.
And if you do get that far you'll have seen something very
important about mathematics. A monkey eating peaches might seem a
rather childish puzzle, but once we start to generalize the problem
we find a valuable abstract form which will appear in the most
A mathematician's task
to have a good stock of these abstract forms and to know their key
Mathematicians are always looking for new forms and new properties;
posing questions around simple situations is a great way to make