No
51
Zigzag
Numbers
Can you arrange the numbers 1, 2, 3
and 4 so that they alternatively rise and then fall? There are five
different ways to do this:
|
|
Ends in 1
|
Ends in 2
|
Ends in 3
|
Ends in 4
|
|
|
-
|
3412
|
1423
|
1324
|
|
|
|
|
2413
|
2314
|
|
Totals:
|
0
|
1
|
2
|
2
|
The situation gets even more
interesting for 1, 2, 3, 4 and 5 with a grand total of 16 different
ways of doing this zigzag arrangement. The following table is not
complete. Can you find the missing arrangements?
|
|
Ends in 1
|
Ends in 2
|
Ends in 3
|
Ends in 4
|
Ends in 5
|
|
|
?
|
14352
|
14253
|
13254
|
-
|
|
|
?
|
?
|
?
|
23154
|
|
|
|
?
|
?
|
?
|
|
|
|
|
?
|
?
|
?
|
|
|
|
|
?
|
?
|
|
|
|
|
Totals:
|
5
|
5
|
4
|
2
|
0
|
Try working systematically finding
the totals for the different zigzag arrangements for 1, then for 1
and 2, then for 1, 2 and 3, then for 1, 2, 3 and 4 etc. You will
eventually discover the rows of the amazing ZIGZAG TRIANGLE:
Can you see how to build up the entries in this triangle
using the cumulative total of the numbers in the previous row? The
alternate numbers down the left hand edge 1, 1, 5, 61... are called
the ZIG numbers and the alternate numbers down the right hand edge 1,
2, 16... are called the ZAG numbers. Why are the ZIG numberws also
called the SECANT numbers and why are the ZAG numbers also called the
TANGENT numbers? Try searching
the Internet to find out. Happy hunting!!
If you have answers, discoveries, new questions etc to do
with this Number Fact then...
- E-mail us at madrascollege.maths@fife.gov.uk
- ...and we will publish them here.
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