Numbers are full of surprises—
andlotsofkookypatterns.
Here’s just one example.
hink of an even number
between0and9.Createanew
number by writing down the
original number, then half this
number, and then the original
number again. Divide the result
bytheoriginalnumber.
Suppose you start with 2.
You would write 212. Divide
this by the original number.
212/2 = 106.
For4,theequationis
424/4 = 106.
What answer do you think
you’llgetstartingwith6or8?Tr y it.
Seven-year-old Dylan
Trueblood,akeenMusereader,
discovered this pattern one
nightwhenhecouldn’tsleep
and started playing with
numbers. And he went further.
What happens when you start
with even numbers that are 10
or larger? hings change a bit.
With 10, you get 10510/10
= 1051. You obtain the same
result with 12. How long does
thisnewpatternlast?When
does the pattern change again?
And again? Get out your
calculator and start exploring.
With help from math professor
José Alberto Macias, Dylan
worked out a formula that
givestheanswerforanyeven
number, no matter how large
thenumberis.Mathematicians are always
looking for patterns and
relationships among numbers.
Here’s one that was discovered in
the 1950s. Start with the sequence
of counting numbers: 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,
and so on. Take out every second
number, keeping only the odd
numbers: 1, 3, 5, 7, 9, 11, 13, 15,
and so on. h en i nd the totals of
the odd numbers in the following
way: start with 1 and add it to the
next number, 3, to get 4. Add the
new total to the next number, 5,
to get 9. Add 9 to 7 to get 16. Add
16 to 9 to get 25, and so on. Write
down the cumulative totals: 1, 4, 9,
16, 25...
What’s special about these
totals? h ey are all squares! In
other words, 1 = 1 x 1, 4 = 2 x 2, 9
= 3 x 3, 16 = 4 x 4, and so on.
A pattern called the
Collatz conjecture has bal ed
mathematicians for decades.
Start with any number. If it is
even, divide it in half; if it is odd,
multiply it by 3 and add 1. Do
the same thing to the result, and
keep on going.
So, starting with 5, you get 3
x 5 + 1 = 16 and then 8, 4, 2, and- From that point on, the “4, 2,
1” pattern keeps repeating. h e
same thing happens with 11. You
get 34, 17, 52, 26, 13, 40, 20, 10, 5,
16, 8, 4, 2, 1, and so on.
Mathematicians have found
that every starting number
they have ever tried eventually
What will you discover when
you play around with math?CHASING NUMBERS
text © 2017 by Ivars PetersonDo
the
Math