Here's the Power Crazy puzzle:
"What can
you say about the values of n that make 7n
+ 3n a multiple of 10?
and
Are there other pairs of integers between 1 and 10 which have similar properties? "
We might begin by using Excel to get some results for 7n + 3n , for integer values of n from 1 upwards, and see which of these are multiples of 10
The Excel file Power Crazy.xls shows this: right-click on the link, choose "Save Target As", and select where you want the file to be saved.
Notice that the spreadsheet doesn't solve the problem, it just does the calculation!
In fact, no spreadsheet, however many results it shows, can be a replacement for the reasoning process.
So what pattern do you see?
And (key question) why does it occur?
Can we now answer the first part of the puzzle:
"What can you say about the values of n that make 7n + 3n a multiple of 10? "
And how about the second part to the puzzle:
"Are there other pairs of integers between 1 and 10 which have similar properties? "
It does seem possible that there may be a connection between the two base numbers in the calculation and the multiple being considered; between the 3, 7, and 10.
So what about other combinations?
Suppose we allowed the base numbers to vary, and showed the results as a table.
Download the Excel file Power Crazy 2.xls
The first result, in C4 , comes from the formula: =B4^M2+C3^M2
Where B4 is the first value in the left-hand column and C3 is the first value
in the top row.
M2 is the index (Power) value, which is adjusted using the increment button
(spinner).
By this formula, B4 and C3 are both raised to the power M2, before being added together.
But the formula is in fact: =$B4^$M$2+C$3^$M$2 , and those dollar signs need explaining.
The purpose of the $ signs is to provide absolute references when the formula is copied from this first cell across the rest of the table.
$B4 has a $ in front of the B so that this cell reference will always come back to column B to collect a value to be used in the calculation.
Similarly C$3 has a $ in front of the 3 so that this cell reference will always come back to row 3 to collect a value to be used in the calculation.
The index (Power) value will always come from M2 so the two $ signs in $M$2 fix both the row and the column in this cell reference.
If you need help making a table of two independent variables check the technique notes from October 02
You may also want to 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.
I have also used Conditional Formatting to pick out the particular values I'm looking for - the multiples of 10.
Conditional Formatting is used to make values of interest conspicuous, without the need for a cell-by-cell inspection, and is explained more fully in July 03 but the conditional formatting formula used here is :=MOD(C4,10)=0
MOD returns the remainder when C4 is divided by 10, and the condition required in this case is that the remainder should be zero.
Excel has done the calculation for us (after we told it what we wanted done), but it has not explained why anything happens (or doesn't happen). That's our job!
Look at what produced multiples of 10. One arrangement group which includes the 3-7 combination, looks very promising.
It is the group, 1-9, 2-8, 3-7, 4-6, and 5-5, which all work provided we use only odd values of n.
It would be nice to generalise and prove something here.
The next bit depends on how much algebra you want to do, but you can take a big leap forward in understanding what's going on if you look at xn + (10-x)n
For example if x is 3, then (10 - x) would be 7 , and we have the original puzzle.
So by using algebra we can analyse the general situation, not just the 3-7 combination. x is the variable allowing us to represent the whole group.
Here's what to do:
Look at xn + (10-x)n for the case when n = 2:
x2 + (10-x)2 can be re-written as 100 - 20x + 2x2 , because (10 - x)2 is the same as 100 - 20x + x2.
This doesn't help very much, but the next n value is more interesting:
xn + (10-x)n for the case when n = 3:
That's x3 + (10-x)3 , and can be re-written
as 1000 - 300x + 30x2 , because (10
- x)3 is the same as 1000 - 300x + 30x2 - x3
and the final term - x3 exactly cancels out x3 the
first term of the whole x3 + (10-x)3 expression.
The terms that then remain will all have a factor of 10, and so the whole expression must be a multiple of 10.
Continuing the algebra for the cases when n = 5, or 7, or 9, and so on, there will always be a final term to exactly cancel out the first term, and all other terms will have a factor of 10.
So what have we explained?
That for all odd n : xn + (10-x)n will always be a multiple of 10
Would odd values of n make 5n + 3n a multiple of 8 and so on?
Here's an Excel file that let's you change the value of the multiple being picked out with conditional formatting: Power Crazy 3.xls
And what about those combinations which produced multiples of 10 using values of n that were not odd? For example: 62 + 82 or 72 + 92
However much of the algebra you use and follow, the purpose of this web page is to illustrate how a spreadsheet can be created to give us numbers, and leave our brains free to think about what we see and why it happens. And like any tool, it takes practice!