Stanford Math Circle
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Stanford Math Circle: Sunday, May 9, 2010
Geometric Numbers
Triangular numbers are numbers that can be arranged in a triangular pattern. Visualize each
triangle as sitting inside the next. The n-th triangular number Tn is formed using an outer triangle
whose sides have n dots:
The first 5 triangular numbers are 1, 3, 6, 10, 15. Observe that the n-th triangular number, which
we will denote Tn , is
n(n + 1)
Tn = 1 + 2 + 3 + · · · + n =
.
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Square numbers are numbers that can be arranged in the shape of a square:
Visualize each square as sitting inside the next. The n-th square number is formed using an outer
square whose sides have n dots. The n-th square number is Sn = n2 .
A pentagonal number is a number that can be arranged in the shape of a pentagon:
The first four pentagonal numbers are 1, 5, 12, 22. Visualize each pentagon as sitting inside the
next one. The n-th pentagonal number is formed using an outer pentagon whose sides have n dots.
Hexagonal (and septagonal, r-gonal, etc.) numbers are defined similarly.
Note: Whenever possible, try to come up with geometric (rather than induction) proofs of the
properties in the following problems.
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Geometric Numbers
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1. Explain geometrically why Tn =
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n(n + 1)
for all integers n ≥ 1.
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2. (a) Compute T1 + T2 , T2 + T3 , T3 + T4 , and T4 + T5 .
(b) Computing more expressions of the form Tn + Tn+1 if necessary, make and prove a
conjecture about the sum of any two consecutive triangular numbers Tn and Tn+1 .
3. (a) Compute 8T1 + 1, 8T2 + 1, 8T3 + 1, 8T4 + 1, and 8T5 + 1.
(b) Computing more expressions of the form 8Tn +1 if necessary, make and prove a conjecture
about the value of 8Tn + 1.
4. Can you find any interesting equations that relate Ta+b To Ta and Tb ? How about Tab ?
5. Show that if T is a triangular number, then 9T + 1 is also a triangular number.
6. What are the possible digits that a triangular number can end in?
7. What are the possible digits that a square number can end in?
8. What are the possible last 2 digits that a square number can end in?
9. The digital root of a number is obtained in the following way. Start with your number, and
sum its digits. Then sum the digits of the resulting number, and continue until only one digit
remains. This is called the digital root. What are the possibilities for the digital root of a
triangular number? What are the possibilities for the digital root of a square number?
10. How many four digit square numbers are composed of only even digits? What four digit
square numbers can be reversed and become the square of another number?
11. (a) Compute 13 , 13 + 23 , 13 + 23 + 33 , 13 + 23 + 33 + 43 , and 13 + 23 + 33 + 43 + 53 .
(b) Computing more expressions of the form
13 + 23 + 33 + · · · + n3
if necessary, make a conjecture about how the sum of the first n cubes is related to the
n-th triangular number Tn . Prove that your conjecture is correct for all integers n ≥ 1.
12. (a) Compute 3T2 + T1 , 3T3 + T2 , 3T4 + T3 , 3T5 + T4 , and 3T6 + T5 .
(b) Computing more expressions of the form 3Tn +Tn−1 if necessary, make a conjecture about
the expression 3Tn +Tn−1 , and prove that your conjecture is correct for all integers n ≥ 1.
13. Can you find any triangular numbers whose square is also a triangular number?
14. Compile some data and try to make a conjecture about which numbers can be written as a
sum of two triangular numbers. For example,
7 = 1 + 6 and 25 = 10 + 15
are sums of two triangular numbers, while 19 cannot be written as the sum of two triangular
numbers. Can you prove your conjecture?
15. There are 6 triangular numbers that can be expressed as the product of three consecutive
integers. Can you find them?
16. Triangular numbers that can be expressed as a product of two primes are called triangular
semiprimes. For example, 6 is a triangular semiprime because 6 = 2 · 3. Can you find other
triangular semiprimes?
17. Are there 4 distinct triangular numbers in geometric progression?
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18. Show that every even perfect number is triangular. Perfect numbers are numbers n with the
property that the sum of the proper divisors of n (not including n) sum to n. For example,
6 is a perfect number because 1 + 2 + 3 = 6.
19. (?) Show that every positive integer can be expressed as a sum of 3 or fewer triangular
numbers.
20. Investigate the minimum number of squares needed to represent a given number. Do you
see any patterns? For each number k, compare the minimum number of squares needed to
represent k with the minimum number needed to represent k 3 . What do you observe? (Note:
it is known that every positive integer can be expressed as a sum of 4 or fewer square numbers.)
21. (a) What is the 5-th pentagonal number?
(b) Find a simple formula for the n-th pentagonal number Pn .
(c) How do pentagonal numbers relate to triangular numbers? Find a number c such that
the following is true: If P is a pentagonal number, then there is a triangular number T
such that P = cT .
(d) There are conjectured to be exactly 210 positive integers that cannot be expressed as
the sum of 3 pentagonal numbers. Find 6 of them.
(e) There are only 6 positive integers that cannot be expressed as the sum of 4 pentagonal
numbers. Find them.
(f) (?) Show that every positive integer can be expressed as a sum of 5 or fewer pentagonal
numbers.
22. (a) Find a general formula for the n-th hexagonal number.
(b) Show that every hexagonal number is also a triangular number. Is every triangular
number also a hexagonal number? If not, can you classify which ones are?
(c) There are exactly 13 positive integers that cannot be expressed as a sum of 4 hexagonal
numbers. Find 6 of them.
(d) There are only 2 positive integers that cannot be expressed as a sum of 5 hexagonal
numbers. Find them.
(e) (?) Show that every positive integer can be expressed as a sum of 6 hexagonal numbers.
23. More generally, find a formula for the n-th r-gonal number. Show that every positive integer
can be expressed as a sum of r r-gonal numbers.
24. A tetrahedral number is a number corresponding to a configuration of points that form a
pyramid with a triangular base:
(a) What are the first 5 tetrahedral numbers?
(b) Find a general formula for the n-th tetrahedral number.
(c) How does the n-th tetrahedral number relate to Pascal’s triangle?
(d) Are there any numbers that are both triangular and tetrahedral?
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(e) Are there any numbers that are both square and tetrahedral?
(f) Pollock’s Conjecture (1850) states that every number is the sum of at most 5 tetrahedral
numbers; this conjecture has not yet been proven. It is also conjectured that there are
exactly 241 numbers that cannot be written as the sum of 4 or fewer tetrahedral numbers.
Can you find the first 5?
(g) How would you define a square pyramidal number? A pentagonal pyramidal number?
A hexagonal pyramidal number? Once you’ve defined a square pyramidal number, show
that the sum of two consecutive tetrahedral numbers is a square pyramidal number.
This is, of course, analogous to the 2-dimensional result (the sum of two consecutive
triangular numbers is a square number).
25. The centered polygonal numbers are numbers formed by a central dot, surrounded by
polygonal numbers with a constant number of sides. Each side of a polygonal layer contains
one dot more than a side in the previous layer.
(a) Find the first 5 centered triangular, centered square, centered pentagonal, and centered
hexagonal numbers.
(b) Find a general formula for the n-th centered k-gonal number. Can you explain your
formula geometrically?
(c) How would you define a centered cube number? Can you find a general formula for the
n-th centered cube number?
26. Investigate formulas for and properties of other geometric numbers. For example, a rhombic
dodecahedral number is a number constructed as a centered cube with a square pyramid
appended to each face.
What is an octahedral number? How do octahedral numbers relate to pyramidal numbers?
What is Pollock’s conjecture for octahedral numbers?
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Geometric Numbers