Challenge Level

I have an unlimited supply of planks, of lengths 7 and 9 units. Putting planks end to end, what total lengths can be achieved?

For example, I can make a length of 30 by using three planks of length 7 units and one of length 9 units, but what about a total length of 40?

The Excel file Planks79.xls shows all the possible combinations: I usually right-click on the link, choose "Save Target As", and select where I want the file to be saved.

In one direction (down) are the multiples of 7, including 0, and in the other direction (across) are the multiples of 9. The numbers in the table are the totals achieved with each combination.

If you need help making a table of two independent variables check the technique notes from October 02

So, is there a value after which all totals must be possible?
And if so, was that value predictable?

Notice that the spreadsheet doesn't solve the problem, it just
does the calculation - instantly!

If you think there is a value after which all totals must be
possible, you will need to reason beyond the numbers that happen to
be showing in the spreadsheet.

In fact, no spreadsheet, however many results it shows, can be
a replacement for the reasoning process.

Don't move on too quickly. Play lots with the 7 and 9
situation, but when you're ready try to generalise :

Can you predict what can and cannot be done for any pair of
plank lengths?

The Excel file Planks.xls is similar in
structure to Planks79 but the plank lengths are now variable.

Look back at October 02 to find out about
increment buttons. They're very easy and let you keep your
attention on the numbers instead of the keyboard.

Try to generalise the investigation to cover any combination
of plank lengths. Is there always a certain value, after which all
totals must be possible? Can you say what that certain value will
be?

Suppose the planks were 6 and 9. Remember Excel can do the
calculation for you (provided you can say what you want done), but
it cannot explain why anything happens (or doesn't happen).

I have also used Conditional Formatting to pick out the
particular values I'm looking for at any time. Conditional
Formatting is used to make values of interest conspicuous, without
the effort of a cell-by-cell inspection. It is explained more fully
in the techniques section.