When a five-digit number, with "1" as the thousands digit, is multiplied
by 4 1 7 the result is a seven-digit number which begins with 9 and ends 0
5 7
Find the missing digits for both these numbers.
Here's the Excel file: X marks the Spot.xls ( Right-click on the link, "Save Target As", and select where you want the file to be saved )
The increment buttons ( called spinners in Excel ) change the values of the digits, and automatically produce new results for the multiplication.
The answer to the puzzle is 2 1 9 2 1 but a class are very unlikely to get that just by flicking though digit values.
Many of us tackle this kind of puzzle with a mixture of reasoning and trial and error (or trial and improvement) :
but even when we hit upon a solution by trial and error, it's good to ask whether we can now see a reason why that had to be the answer.
This activity could just be a warm-up, to draw a class into some mathematical
thinking at the start of the lesson.
Or it could be the first example of a more open investigation based on place
value.
If I did want to make this a more extended activity I might begin with the " X marks the Spot " puzzle as an example, let the class together try it and discuss it, then invite children to create a puzzle like this for themselves, but to start with something simpler :
Multiply two numbers, hide a few digits, then swap with a friend. Try each other's puzzle, check and discuss together, then share puzzles with other pairs.
There's lots to discuss, of course, but I would want to include consideration
of puzzles that have more than one solution.
If none arise naturally from the class, I'd ask them to find the most digits
that can be removed until a solution is no longer unique.
Differentiation by challenge is easy enough to achieve.
I can control the size of the numbers that children use, and also vary the
balance between the use of reasoning and trial and error.
For help with Spinners, look back at October 02 , increment buttons are very easy to insert and keep attention on the changing numbers, instead of the keyboard.
I have used Conditional Formatting to create a colour change ( to bright red
) when the right answer occurs.
And have also used Conditional Formatting to conceal the ten-millions digit
if it's zero ( font colour matched to background ) , and also to turn the border
on or off for that cell . Conditional Formatting is explained more fully in
the technical notes for July
03
Cell K6 does the multiplication, before the digit boxes each receive their correct value from the digits in that result.
Although column & row headers have been turned off in Tools/Options/View , the location reference for the current cell can still be seen beside the Formula bar, immediately above the worksheet.
The individual digits are isolated using the INT function. INT takes just the integer part of a value and ignores the rest, for example INT of 3.845 is just 3.
Using G2 as an example, this cell calculates the hundreds digit with the formula : = INT ( $K$6 / 100 ) - 10 * INT ( $K$6 / 1000 )
K6 was only made an absolute reference, $K$6 , to help me copy a basic formula across all the cells for digits and then just adjust the number of zeros required in each formula.
INT ( K6 / 100 ) calculates the number of whole hundreds in the multiplication product.
And from this I need to subtract 10 times the number of whole thousands in the product
This technique of isolating digits was also used in the " Happy Numbers " investigation which appeared in Febuary 02