


This is a small sampling from
my collection of over 400 patterns. I hope you enjoy them. And
please call again soon.
Thanks to Dr. M. Ecker for adding the 9 <> 0 above.
See a picture of the above pattern in crossstitch, on my models
page. 
2010 Note: Because this section was a prominent part of the original
web site, I have chosen to include it on this renamed Magic Hypercubes
site.

A brief explanation of what this page is all about. 

The sum of N odd integers starting at 1 equals N
squared. 

The sum of N odd integers starting after the
previous series equals N cubed. 

4^{2}, 34^{2}, 334^{2}, etc.
and three other similar numbers. 

1089 x 9, 10989 x 9, etc., & their reversals x 4. 

New patterns added July 16, 2007 
Other pages in this section



A page with 45+ prime number patterns. 

Prime plots, Prime Queens problem, Prime rectangles,
etc. 

Patterns where the number equals some manipulation
of its digits. 

Patterns where the number equals some manipulation
of its digits. 

Introduction
Herein are presented a miscellany of number patterns and interesting numbers.
They are displayed with very little commentary but with the hope they will
increase the viewer's appreciation for the beauty that can be seen in
mathematics.

NOTE for elementary mathematics teachers; perhaps this
material will be of value to your students for mathematics enrichment.
Recreational mathematics hobbyists. Can you add
material or insight to these patterns? 
See my policy on this web site regarding credits at
All
About Credits.
Relation of
consecutive odd numbers to squares.
The sum of N odd integers
starting at 1 equals N squared.
Relation of consecutive
odd numbers to cubes
The sum of N odd integers
starting after the previous series equals N cubed.
Four Related Patterns
1089 and it's
reversal
These are the only 4, 5, 6 and 7 digit numbers that are equal to a multiple of
their reversal.
(The series may be extended indefinitely by inserting nines into the center of
each number.)
See more on 1089 and the LhoShu magic square at
SquaresUpdate.htm
17 = 2^{3}
+ 3^{2}
This is the only number
of the form x + p^{q} + q^{p} .
17 = 3^{4}
– 4^{3}
Is this the only number
of the form x + p^{q}  q^{p} ?
JRM 30:3, page 234. Problem 2447
proposed by Michael Ried.
1729 = 10^{3}
+ 9^{3} = 12^{3} + 1^{3}
Ramanujan observed that this is the smallest number with this property.
Ramanujan’s number plus the number of the beast equals the sum of the first
prime and the squares of the next 9 primes.
1729 +
666 = 2 + 3^{2} + 5^{2} + 7^{2} + 11^{2} + 13^{2}
+ 17^{2} + 19^{2} + 23^{2} + 29^{2} = 2395
Chanchal Singh, JRM 21:2, page 135. He also
shows other facts about 1729.
27 x 594 =
16038
This is the only example of the 10 digits appearing in numbers
of 2, 3, and 5 digits in this form.
Also, 594 is also an exact multiple of 27.
The Canterbury Puzzles, H. E.
Dudeney, Dover Publ., 1958, Puzzle 101, Page 242
3816547290
is the only base
10 number that…..
 uses each of
the 10 digits exactly once
 the leftmost
k digits are evenly divisible by k.
3
is evenly divisible by 1
38 is evenly divisible by 2
381 is evenly divisible by 3
…
3816547290 is evenly divisible by 10.
Mike Kieth, REC #91
(vol.13, No.5), Jan. 1999, page 11. 
3608528850368400786036725 This is the largest
integer that has the property that it’s
leftmost k integers are evenly divisible by k (in this case 1 to
25).
Mike Kieth via email Jan.
20, 1999. 
(15^{2}
– 15^{1 }/ 15^{1}) + 15^{0} = 15
This works for any
number.

2 x
9 = 18
and
9^{2}
= 81 
1/243 = 004113226337448559 
124983/576 = 216.984375
Digits 1 to 9
on each side of the equation. 
2340819/576 = 4063.921875
Digits 0 to 9
on each side of the equation. 
9^{1}
= 9
9^{2} = 81
and
8 + 1 = 9
9^{3} = 729
and
7 + 2 + 9 = 18
and
1 + 8 = 9
9^{4} = 6561
and
6 + 5 + 6 + 1 = 18
and
1 + 8 = 9
9^{5} = 59049
and
5 + 9 + 0 + 4 + 9 = 27
and
2 + 7 = 9
This pattern continues
indefinitely .
This pattern looks good, but is the result of the simple fact that all multiples
of 9 have a digital root of 9.
Matt Medlock (via
email Nov. 21, 2002
(1 + 5
+ 4 + 7)(1^{2} + 5^{2} + 4^{2} + 7^{2}) = 1547
(2 + 1 + 9 + 6)(2^{2} + 1^{2} + 9^{2} + 6^{2}) =
2196
Other numbers
with this property are 1, 133, 315, 803, and 1148.
Aktar Yalcin (via
email June 25, 2007
