2016 - some of the more unusual, fascinating and fun facts about the number.
Some are left as puzzles for you to solve with a to show the solutions.
The facts below are all derived from Neil Sloane's On-line Encyclopedia of Integer Sequences
where the sequences containing 2016 listed here are shown as, for instance, A224950
which is also a link to the series itself.
Contents of this page
First some fun facts about the year 2016 and then about the number 2016.
The year 2016
2016 is a leap year with 53 Fridays and 53 Saturdays. A224950
There are 5 Mondays in February this year.
The last time that happened was 1988 and the next is 2044. A135795
The transit of Mercury across the sun will be visible from somewhere on the earth this year.
The next time is in 2019. A171466.
The number 2016
What 2016 looks like
Small:
Large:
Very Large:
If we draw 64 points round a circle then join each to every other, there will be 64×63/2 = 2016 lines.
Here it is and it must surely be the prettiest way to see 2016!
Fun ways to write 2016
Alex Bellos in his Monday Puzzle of 4 January 2016
in the Guardian newspaper posed the problem of using the numbers 1 to 10 and various maths operations
to make 2016. Surely the most elegant (to quote Alex and I agree) is the following from James Annan and others:
10×9×8×7×6
= 2016
5+4+3+2+1
It turns out that this is just one of a whole pattern where we add the first n numbers on the bottom and multiply the next n on the top.
2
= 2,
4×3
= 4,
6×5×4
= 20, ...
2n×(2n-1)×...×(n+2)×(n+1)
1
2+1
3+2+1
n+(n-1)+...+2+1
It is an easy exercise
to show that they will all be whole numbers. A110371
In different Bases
In binary (base 2) 2016 is 11111100000 with six 1s followed by five 0s.
The next number with this pattern of a block of 1s followed by a block of 0s is 2032 which is 11111110000 in binary. A006516,
A043569
Also, the digit sum of the binary number 11111100000 is 6 which is a factor of 2016.
If we write 2016 in every base from 2 to 10 and sum the digits each time, we will always get a total that is
itself a divisor of 2016.
Base
2016 in base
Digitsum
2016/Digitsum
2
11111100000
6
336
3
2202200
8
252
4
133200
9
224
5
31031
8
252
6
13200
6
336
7
5610
12
168
8
3740
14
144
9
2680
16
126
10
2016
9
224
We can go beyond base 10 if we represent the digits in bases beyond 10 as their decimal values
for example, 2016 = 11×132 + 12×13 + 1 = 11 12 1 then
this property of 2016 extends for several more bases yet. How many?
Base
2016 in base
Digitsum
2016/Digitsum
11
1 5 7 3
16
126
12
1 2 0 0
3
672
13
11 12 1
24
84
14
10 4 0
14
144
15
8 14 6
28
72
16
7 14 0
21
96
17
6 16 10
32
63
For base 18, 201618=6 4 0 with digitsum 10 but 2016 is not a multiple of 10.
How many ways can you write 32 as a sum of 7 squares if 0 and repetitions are allowed and also
each different order of the 7 squares counts as a different answer?
There are a total of 2016 solutions!
That is too many to list
by hand so try this simpler problem :
find the 6 ways of writing 32 as a sum of 7 squares, 0 and repetitions being allowed, if the
numbers to be squared are written in order of size. For example,
02 + 02 + 02 + 02 + 02 + 42 + 42 = 32
but we do not count
02 + 02 + 02 + 02 + 42 + 02 + 42 = 32
because 0,0,0,0,4,0,4 is not in the correct order.
Can you find 4 whole numbers a, b, c and d, all in arithmetic progression (when written in order each differs
from the next by the same amount) and the sum of their squares is 2016?
A Runsum is a sum of a run of consecutive integers:
2016 = 1 + 2 + 3 + ... + 63
2016 = 86 + 87 + ... + 106
2016 = 220 + 221 + ... + 228
2016 = 285 + 286 + ... + 291
2016 = 671 + 672 + 673 Runsums
Factors
2016 = 2×2×2×2×2×3×3×7 = 32×63 = 26−1(26−1)
2016 = 16 ×126: 2016 is the smallest multiple of 16 (after 16 itself) that ends 016
What is the next multiple of 16 ending 016?
4016 = 251 × 16
2016 = 8 × 252
If we reverse the digits in 8 and 252, we get the same numbers. But also
2016 = 24 × 84 = 42 × 48
where if both factors are again reversed the new numbers still have the same product 2016!
So 2016 has two separate pairs of factors that, when the individual factors are reversed, they still have a product of 2016.
What is the smallest number with two pairs of factors like this?
What is the next such number after 2016?
The smallest number with two such pairs of factors is 484 and the next after 2016 is 2178:
484 = 11×44 = 22×22
2178 = 22×99 = 33×66
See also A262873.
If we exclude the self-reversing factor pairs such as 8 × 252 which give the same pair when they are reversed,
then see A228164
If we take a list of the numbers from 1 to 4, we can find 16 subsets of them:
{}, {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}
Take each subset and multiply all the numbers in the subset. Some products will
be repeated but there are only 8 different products.
If we try this with the numbers from 1 to 14, there are exactly 2016 different products. A060957
215 = 32768 where the product of the digits is 3×2×7×6×8 = 2016. A014257
What is the next year that is a product of the digits of a power of 2?
2700
216 = 65536 and 6×5×5×3×6 = 2700
Pythagorean Triangles
A Pythagorean Triangle is a right-angled triangle whose sides are whole numbers, such as sides 3,4 and 5.
Once we have one such triangle, we can always multiply all the sides by 2 say to get 6-8-10 or by 3 to find 9-12-15
and so on. All these have the same shape.
If we want different shapes then the smallest
Pythagorean triangle of each shape is called a primitive Pythagorean triangle or PPT for short. All
Pythagorean triangles are PPTs or a multiple of a PPT.
There is a whole series of PPTs where the longest side, the hypotenuse, is just 1 more than one of the other sides as here
in the 3-4-5 triangle. The next is 5-12-13, and the series continues 7-24-25, 9-40-41, 11-60-61, 13-84-85, 15-112-113.
If we add up the areas of these 7 triangles we have a total area of 2016.
The Pythagorean triangle 32-126-130 which has sides double those of the PPT 16-63-65
has an area 2016.
You can find exactly 11 integer-sided triangles (not necessarily right-angled) which have an area of 2016.
Six of them are 52-80-84, 48-85-91, 58-70-96, 65-72-119, 32-126-130 (right-angled), 64-225-287.
In 64-225-287, the two sides 64=82 and 225=152 are themselves square numbers.
What are the other five integer-sided triangles with an area of 2016?
In maths we have square numbers 12=1, 22=4, 32=9, ... , each a number multiplied by itself.
But a square number is so called because that number of dots can be arranged in a square shape.
The Square Numbers
side
1
2
3
4
5
6
shape
size
1
4
9
16
25
36
Other shapes are triangular numbers 1, 1+2=3, 1+2+3=6,
If we take a board of squares arranged in a pyramid shape, we can place a domino on it either
horizontally:
or vertically:
to cover two squares on the board.
The first board here has 4 squares on its base. There are just 6 ways to place a domino on the squares:
4 with it horizontal and 2 verticallly.
If the pyramid board has a base of 32 squares and is 16 levels tall, we can place a domino on it in 2016 ways. A000384
A Magic Square of Prime Numbers
6
1
8
15
7
5
3
15
2
9
4
15
15↗
15
15
15
↖15
79
137
197
199
277
347
349
431
127
193
131
419
337
421
107
281
103
379
283
389
293
227
179
163
397
251
83
271
269
157
439
149
409
211
383
191
181
101
401
139
307
239
317
167
89
367
97
433
353
233
359
151
257
223
331
109
241
373
263
229
313
173
113
311
A Magic square is a square of whole numbers with every row, column
and both diagonals having the same sum, called the magic constant for the square.
On the left is a magic square using the numbers 1 to 9 and each row, column and diagonal sums to 15.
We can use all 64 consecutive prime numbers between 79 and 439 to make an
8x8 magic square with magic constant 2016. This is the smallest magic constant in an 8x8 square of
consecutive primes.
A073520 A189188
Found by G Abe and A Suzuki, two "amateur" mathematicians and
reported in The Study of Magic Squares G Abe, 1957 (Japanese only).
See prime puzzles and
problems page.
More on Squares
Painting walls
In a 9x7 cm rectangle of squares, if we trace around every square of every size
then we would have drawn a line 2016 cms long.
There are 63 of size 1x1, some of size 2x2, and so on, up to the three of size 7x7.
For each size from 1x1 to 7x7, calculate how many squares there are of that size
and therefore how long the perimeter is of all the squares of each size and so verify that the total is 2016.
There are
9×7
=
63
squares of size
1×1
each with perimeter
4×1
=
4
cm
There are
8×6
=
48
squares of size
2×2
each with perimeter
4×2
=
8
cm
There are
7×5
=
35
squares of size
3×3
each with perimeter
4×3
=
12
cm
There are
6×4
=
24
squares of size
4×4
each with perimeter
4×4
=
16
cm
There are
5×3
=
15
squares of size
5×5
each with perimeter
4×5
=
20
cm
There are
4×2
=
12
squares of size
6×6
each with perimeter
4×6
=
24
cm
There are
3×1
=
3
squares of size
7×7
each with perimeter
4×7
=
28
cm
That's a total of 63×4+48×8+35×12+24×16+15×20+12×24+3×28 = 2016 cm.
If we draw a circle of radius 4 on graph paper, we find there are 32 complete squares inside the circle.
There are 2016 squares completely inside a circle of radius of 26. A119677 are the alternate terms of
A136485. Also see
A261849.
Hypercubes
The 9-dimensional hypercube of 512 vertices and 2304 edges has 2016 5-dimensional hypercubes (penteracts). A135273
Hypercubes are the continuation of the series:
a point, a line, a square, a cube into 4 and more dimensions.
Each is derived from the previous one by taking a copy of it but imagining that all
the points leave a trail from the original to the copy, forming new edges in the new dimension.
Shape
∙
Dimension
0
1
2
3
4
The number of points will double each time we go to the next dimension.
The number of edges will double but also there are extra edges between the pairs points between the original and the copy.
The square faces and cubes are more difficult to see but the formula is as follows:
we have taken two copies of the previous dimensional object so we double the number of faces (cubes, ...) etc.
But we also have made new edges, faces, etc equal to the number of 1-dimension-smaller objects, so the extra
faces are made from each edge in the smaller dimension, the extra cubes are made form each face in the smaller dimension
and so on.
So from a cube with 8 points and 6 square faces and 12 edges to the next dimension, we find the faces are
2×6 faces and 12 square faces from the 12 edges: a total of 24 square faces in the 4D hypercube.
Similarly, the number of edges in the 4D hypercube cube is twice 12 edges from the two cubes plus 8
more where the 8 points of the cube now become edge in the 4D version: a total of 24+8=32 edges.
A Pythagorean Triangle is a right-angled triangle whose sides are whole numbers, such as sides 3,4 and 5.
Dr Ron Knott
