Download Cool Math Newsletter

WHOA! COOL MATH
CURIOUS MATHEMATICS FOR FUN AND JOY
NOVEMBER 2012
PROMOTIONAL CORNER: Have
you an event, a workshop, a website,
some materials you would like to share
with the world? Let me know! If the work
is about deep and joyous and real
mathematical doing I would be delighted
to mention it here.
The square numbers arise from
arranging pebbles into square arrays:
and the triangular numbers from making
triangular arrays:
***
Looking for high-end, but completely
accessible, rich, relevant, and
meaningful explorative experiences for
your students? Would you like it all
coupled with guidance for teachers too?
Check out:
MAKING MATHEMATICS:
Mentored Research Projects for
Young Mathematicians.
http://www2.edc.org/makingmath/mathp
roj.asp
You and your students are in for a treat!
This month’s essay simply asks:
Are there more square numbers than
triangular numbers? Are there more
triangular numbers than square
numbers? Or are the counts of each
essentially the same?
Given that there are infinitely many of
each type of number is this question
meaningless? Hmm.
© James Tanton 2012
ASIDE: Some interplay between
square and triangular numbers.
The following diagram shows that
doubling any triangular number and
adding to it its matching square number
gives another triangular number:
For example double 21 , the sixth
triangular number, plus the sixth square
number gives:
2 × 21 + 36 = 78 ,
the twelfth triangular number.
This picture
FORMULAS: Let S N denote the N th
square number and TN the N th
triangular number. We have:
SN = N 2
TN = 1 + 2 + 3 + ⋯ + N
This second formula is not particularly
helpful. But noting that two triangles
placed together make an oblong allows
us to see that TN is half of N ( N + 1)
We have: TN =
N ( N + 1)
2
.
Puzzler: Show, both algebraically and
pictorially, that the triangular numbers
satisfy:
TM + N = TM + TN + MN .
shows that the sum of two consecutive
triangular numbers gives a square. For
instance, 3 + 6 = 9 , 6 + 10 = 16 and
10 + 15 = 25 . (Why don’t two triangles
the same size make a square?)
Puzzler: Can you draw pictures to
explain the following curiosities?
Pick any three consecutive triangular
numbers, multiply the middle one by six
and add to this product the remaining
two. The result is square number.
Show, both algebraically and pictorially,
that the square numbers satisfy:
S M + N = S M + S N + 2 MN .
HIGHER FIGURATE NUMBERS:
The triangular numbers are formed
successively adding rows to the
following diagram:
For instance: 3 + 6 × 6 + 10 = 49 .
Pick any triangular number and multiply
it by eight. The result is always one less
than a square number.
For instance: 8 × 15 + 1 = 121 .
www.jamestanton.com
© James Tanton 2012
The “pentagonal numbers” are formed
by adjoining three diagrams and
successively adding rows:
COUNTING FIGURATE
NUMBERS: Clearly there are infinitely
many square numbers and infinitely
many triangular numbers, and so asking
which there are “more of” can, at first,
seem meaningless. However, if we think
of “more” as a question of “density”
then there might be a difference between
the two worth exploring. For instance
there are 31 square numbers and 44
triangular numbers among the numbers 1
through 1000. These counts are
different. In fact, the following table
suggests there is something interesting
afoot. (The number 1.414 is suspiciously
like the square root of two!)
And so on!
Let’s explore “more of” in terms of
density.
The square numbers are formed by
adjoining two of these diagrams and
successively adding rows:
Find a formula for the N th pentagonal
number PN . Find a formula for the N th
hexagonal number H N . Find a formula
for the N th septagonal number GN .
Show that:
PM + N = PM + PN + 3MN
H M + N = H M + H N + 4 MN
GM + N = GM + GN + 5MN
Question One: If N is a large number,
how many numbers among 1, 2,3,..., N
are square?
The square numbers in this list are the
numbers 1, 4,9,.., k 2 with k the largest
integer satisfying k 2 ≤ N . That is, k is
the largest integer ≤ N . That is, k is
the value N rounded down to an
integer. Mathematicians denote this
 N  (and call it the floor of N ). We


have: There are  N  square numbers
among 1, 2,3,..., N .
www.jamestanton.com
© James Tanton 2012
Question Two: How many numbers
among 1, 2,3,..., N are triangular?
The triangular numbers we seek are
k ( k + 1)
1,3, 6,...,
with k as large as
2
1
possible satisfying k ( k + 1) ≤ N . So
2
we need to find the largest value k
satisfying k 2 + k ≤ 2 N . Completing the
square, we need the largest value of k
with:
1
1
k 2 + k + ≤ 2N +
4
4
2
1
1

 k +  ≤ 2N +
2
4

1 1
−
4 2
This shows: There are

1 1
 2 N + −  triangular numbers
4 2

among 1, 2,3,… , N .
k ≤ 2N +
If N is indeed truly large, adding a
quarter before taking a square root, will
have little effect (the square roots of
2000000 and 2000000.25 are very
close) and subtracting a half from a large
answer will offer negligible change.

1 1
Thus  2 N + −  is very close to the
4 2

number  2N  , which looks about 2
as big as the formula  N  . We see
that the count of triangular numbers is
indeed larger than the count of square
numbers by a factor close to 2 (and
the error of this claim likely becomes
negligible for larger and larger values of
N ). In this sense we can say:
There are 2 more triangular
numbers than square numbers!
Very technical aside: To make this
claim iron-clad we need to establish:

1 1
 2N + − 
4 2
= 2.
lim N →∞ 
 N


This can be done by noting that for any
value x we have: x − 1 ≤  x  ≤ x . Thus
1 3  2 N + 1 − 1 
−
4 2
4 2≤
≤
 N
N


2N +
1 1
−
4 2
N −1
2N +
That is:

1 1
 2N + − 
4 2
1
3
−
≤
≤
2+
4N 2 N
 N


1
1
−
4N 2 N
1
1−
N
2+
As N → ∞ , the left side approaches
2 + 0 − 0 = 2 and the right side
2+0 −0
= 2 . The
1− 0
quantity in between, the ratio we seek,
thus approaches the value 2 as well.
(Ahhh. The “squeeze theorem” from
calculus!)
approaches
RESEARCH CORNER 1:
How many more triangular numbers are
there than pentagonal numbers? Than
hexagonal numbers? Than septagonals?
EXTRA: THE SQUANGULARS!
Here are the square and triangular
numbers:
Notice that 1 and 36 belong to both lists
– they are both square and triangular. I
call these numbers squangular. The next
squangular number is 1225, and then
comes 41616. In fact there are infinitely
many squangular numbers and the N th
squangular number is given by the crazy
formula:
www.jamestanton.com
© James Tanton 2012
(
) (
 3+ 8 N − 3− 8


2 8


)
N





2
[See www.jamestanton.com/?p=519 for
an accessible derivation of this!]
RESEARCH CORNER 2:
Are there any pent-squangular numbers
(apart from 1)? Is there a non-trivial
number that is simultaneously square,
triangular and pentagonal? [I personally
do not know the answer to this.]
numbers halfway between being nonsquare and non-triangular - whatever
that means!
Putting N = 1, N = 2, N = 3,... into
N+
1.5 N
gives the sequence of
“non-halfway-square/triangulars”
2, 4,5, 6,8,9,10,11,13,14,15,16,17,19, 20,...
So the “actual halfway square/
triangulars” are the numbers not listed
here, namely:
1,3, 7,12,18, 25, 33, 42, 52, 63,...
SOMETHING BIZARRE!
Here are the non-square numbers,
everything but the squares:
[This truly is absurd!]
RESEARCH CORNER 3:
It looks like the sequence
2,3, 5, 6, 7,8,10,11,12,13,14,15,17,... .
1,3, 7,12,18, 25, 33, 42, 52, 63,...
The N th number in the list is given by
the formula N +
N . Here the angled
brackets mean round up or down to the
nearest integer.
Challenge: Verify this formula for the
non-squares. (Hint: We know that there
are  N  squares among the numbers
1 through N , so the N th non-square is
the number N “bumped up”  N 
places. But we need careful and check if
this bumping goes beyond yet another
square number.)
Here are the non-triangular numbers:
2, 4, 5, 7,8, 9,11,12,13,14,16,17,....
The N th one in the list is given by the
formula N +
2N .
Challenge: Verify this!
has the property that every counting
number either appears in the sequence or
as the difference of two consecutive
terms of the sequence. (The sequence of
differences is back to 2, 4, 5, 6,8, 9,10,...
by the way!)
Is this actually the case?
RESEARCH CORNER 4:
Is there any meaningful way to actually
interpret the sequence of numbers
1,3, 7,12,18, 25, 33, 42, 52, 63,...
as halfway, in some manner, between the
sequence of triangular numbers and the
sequence of square numbers? [I haven’t
seen any reason for it myself. If you do,
let me know!]
RESEARCH CORNER 5:
Dare I ask about non-pentagonals, or
sequences five-twelfths of the way
between the squares and the
septagonals? (Oh my!)
This suggests that the formula
N+
1.5 N
should correspond to
© 2012 James Tanton
[email protected]
www.jamestanton.com