This is part of live presentation series called Mathematical Games in which we explore a variety of games and puzzles using Wolfram Language. In this episode, we explore the mathematical games and puzzles involving numbers and OEIS.

demonstrations.wolfram.com

Many Demonstrations involve numbers

Welcome to 2025. The Partridge Problem

Happy 2025 =
2
T
9
= 1³+2³+3³+4³+5³+6³+7³+8³+9³ !
On the third day of Christmas my true love sent to me
Four calling birds
Three French hens,
Two turtle doves,
And a partridge in a pear tree.
Each year, we try to list properties for the number of the given year. For 2025, we start by squaring the ninth triangular number, 45. By slicing the cubes into height 1 squares, we get a tiled solution.
Out[]=
2025 =
3
1
+
3
2
+
3
3
+
3
4
+
3
5
+
3
6
+
3
7
+
3
8
+
3
9
1+2+3+4+5+6+7+8+9
2
)
This comes from the following identify, often referred to as Faulhaber’s Formula or Nicomachus’s Theorem:
In[]:=
TraditionalForm[HoldForm[Sum[k^3,{k,1,n}]=(Sum[k,{k,1,n}])^2]]
n
∑
k=1
3
k
=
2

n
∑
k=1
k
We can also do it with 2025 triangles. William Marshall was the first to find that equilateral triangles are order 9. Squares are order 8.

More on 2025

It’s the product of the proper divisors of 45:
The radius of the circumcircle for 8 different integer triangles is sqrt(2025/11):
An odd property of the square root, 45, is that it’s the smallest integer where the periodic part of the reciprocal is 2:
It’s the denominator for this sum of squares:
We can also represent 2025 with ϕ, the golden ratio:
There are also 2025 “good” permutations (A006717), where all rotations have a single number in the correct place.
Here’s the first of the “good” permutations:
It’s the product of greatest common divisors of 15 for numbers less than 15:
It’s the sum of two high powers:
It’s a self-descriptive square:
Note the denominator here:

Graph Properties equaling 2025

Here are various graphs and graph properties that happen to be 2025

We also did this for 2024

2024 is at the end of this
Which leads to this:
2024 is also a dodecahedral number and a tetrahedral number

And for 2023

https://www.wolframalpha.com/input?i=11121217

We can look for a number in Wolfram|Alpha
A266181
Numbers n such that n == d_1 (mod 2), n == d_2 (mod 3), n == d_3 (mod 4) etc., where d_1 d_2 d_3 ... is the decimal expansion of n.

Catalan and OEIS https://oeis.org/A000108

Martin Gardner introduced NJA Sloane’s 1973 Book on Integer Sequences in a Mathematical Games column on the Catalan sequence.
https://oeis.org/A000108 Catalan Numbers
“It was Leonhard Euler who first discovered the Catalan numbers after
asking himself: In how many ways can a fixed convex polygon be divided into
triangles by drawing diagonals that do not intersect?” -- Gardner
There are 2 triangulations for a square, 5 for a pentagon, 14 for a hexagon, 42 for a heptagon, and 132 for an octagon.
The numbers 2, 5, 14, 42, 132,..., are the Catalan numbers.
How many distinct triangulations with rotations excluded? A001683 1, 1, 1, 1, 4, 6, 19, 49, 150, 442
The 49 triangulations of the nonagon might be an interesting card set:

Folding Stamps A001011

1, 1, 2, 5, 14, 38, 120, 353, 1148, 3527, 11622, 36627, 121622, 389560, 1301140, 4215748, 14146335, 46235800, 155741571, 512559195, 1732007938, 5732533570, 19423092113, 64590165281, 219349187968, 732358098471, 2492051377341, 8349072895553, 28459491475593
How many ways can stamps be folded? (The formula for this sequence is unsolved)

Folding a Strip of Unlabeled Stamps (Robert Dickau)

Primes and the Golomb Spiral https://oeis.org/A000040

Ulam’s prime spiral arranges the positive integers in a spiral, marking primes with dark pixels.

Ternary and Balanced Ternary

Common sequences

Square https://oeis.org/A000290

Factorial https://oeis.org/A000142

Triangular https://oeis.org/A000217

Centered Hexagonal https://oeis.org/A003215

5-smooth https://oeis.org/A051037

Perfect, Amicable, Sociable

https://oeis.org/A063990 Amicable

Fibonacci and Padovan Numbers

Armstrong Numbers https://oeis.org/A005188

Numbers that are sums of powers of their digits. The last is 115132219018763992565095597973971522401.

Binary Gray Code https://oeis.org/A003188

The number of Gray Codes on a n-cube is unsolved. https://oeis.org/A003042 .
Consecutive numbers may differ by many bits in an ordinary binary representation of integers; however, they differ by only one bit in the binary Gray code representation. The plots show ranges of numbers starting at 0 whose bits are given in columns counting from the left; the bit differences can be indicated automatically by color.

Crossing Numbers

These are the smallest cubic graphs that require 0, 1, 2, 3, ... crossings.

Polygonal numbers

Pentagonal https://oeis.org/A000326
Hexagonal https://oeis.org/A000384

Wythoff’s Nim

Lower https://oeis.org/A000201
Upper https://oeis.org/A001950

Egyptian Fractions A094871

A sequence by Ahmes, 1650BC. Rhind Papyrus
In ancient Egypt, only fractions with a 1 on top were allowed. Other numbers were sums of these fractions.
How many fractions are needed?

Fusc or Stern’s diatomic series https://oeis.org/A002487

Index of the largest maximal subgroup of PSL(3,q) https://oeis.org/A138077
(7, 13, 21, 31, 57, 73, 91, 133) -- Is this the same as the sequence above?

If a circular wheel is used, a perfect set of measuring marks is called a difference set, a Ganymede circle, or an n-switch. All the distances between selected red points are different. The distances are indicated by arcs inside the circle.

Pancake Cutting https://oeis.org/A000124 Robert Dickau

Unsolved Sequences

Pancake flipping, Sorting by prefix reversal https://oeis.org/A058986

“The chef in our place is sloppy and when he prepares a stack of pancakes they come out all different sizes. Therefore when I deliver them to a customer, on the way to the table I rearrange them (so that the smallest winds up on top and so on, down to the largest at the bottom) by grabbing several from the top and flipping them over, repeating this (varying the number I flip) as many times as necessary. If there are n pancakes, what is the maximum number of flips (as a function a(n) of n) that I will ever have to use to rearrange them?”
There two permutations of order 19 are given which need at least 22 flips.
These permutations are 1,7,5,3,6,4,2,8,14,12,10,13,11,9,15,17,19,16,18 and 1,7,5,3,6,4,2,8,14,12,10,13,11,9,15,18,16,19,17.
What’s the max for Order 20?

Minimal comparisons for a Sort https://oeis.org/A003075
Minimal number of comparisons needed for n-element sorting network.

Distinct Sum Problem https://oeis.org/A276661

Least k such that there is a set S in {1, 2, ..., k} with n elements and the property that each of its subsets has a distinct sum.
a(1) = 1: {1}
a(2) = 2: {1, 2}
a(3) = 4: {1, 2, 4}
a(4) = 7: {3, 5, 6, 7}
a(5) = 13: {3, 6, 11, 12, 13}
a(6) = 24: {11, 17, 20, 22, 23, 24}
a(7) = 44: {20, 31, 37, 40, 42, 43, 44}
a(8) = 84: {40, 60, 71, 77, 80, 82, 83, 84}
a(9) = 161: {77, 117, 137, 148, 154, 157, 159, 160, 161}

Polygon Square Dissection https://oeis.org/A110312
Minimum number of polygonal pieces in a dissection of a regular n-gon into a square

I gave a talk on these. https://community.wolfram.com/groups/-/m/t/2992766

Schur’s numbers https://oeis.org/A045652

no color contains a triple x + y = z
A = {1, 3, 5, 15, 17, 19, 26, 28, 40, 42, 44}
B = {2, 7, 8, 18, 21, 24, 27, 37, 38, 43}
C = {4, 6, 13, 20, 22, 23, 25, 30, 32, 39, 41}
D = {9, 10, 11, 12, 14, 16, 29, 31, 33, 34, 35, 36}
A = {1, 6, 10, 18, 21, 23, 26, 30, 34, 38, 43, 45, 50, 54, 65, 74, 87, 96, 107, 111, 116, 118, 123, 127, 131, 135, 138, 140, 143, 151, 155, 160}
B = {2, 3, 8, 14, 19, 20, 24, 25, 36, 46, 47, 51, 62, 73, 88, 99, 110, 114, 115, 125, 136, 137, 141, 142, 147, 153, 158, 159}
C = {4, 5, 15, 16, 22, 28, 29, 39, 40, 41, 42, 48, 49, 59, 102, 112, 113, 119, 120, 121, 122, 132, 133, 139, 145, 146, 156, 157}
D = {7, 9, 11, 12, 13, 17, 27, 31, 32, 33, 35, 37, 53, 56, 57, 61, 79, 82, 100, 104, 105, 108, 124, 126, 128, 129, 130, 134, 144, 148, 149, 150, 152, 154}
E = {44, 52, 55, 58, 60, 63, 64, 66, 67, 68, 69, 70, 71, 72, 75, 76, 77, 78, 80, 81, 83, 84, 85, 86, 89, 90, 91, 92, 93, 94, 95, 97, 98, 101, 103, 106, 109, 117}
If a+b=c does not exist and all three are distinct, then https://oeis.org/A072842
{ 24 26 27 28 29 30 31 32 33 36 37 38 39 41 42 44 45 46 47 48 49 }
{ 9 10 12 13 14 15 17 18 20 54 55 56 57 58 59 60 61 62 }
{ 1 2 4 8 11 16 22 25 40 43 53 66 }
{ 3 5 6 7 19 21 23 34 35 50 51 52 63 64 65 }
112122213313333333232124144444144422244144441444412223333333331222

No Repeated Distances in a Grid https://oeis.org/A271490

Max length of uncrossing knight path https://oeis.org/A003192

https://oeis.org/A051567 2*floor(2n/3) queens on a nxn board each attacking 1 other.

{0, 5, 0, 2, 149, 49, 1, 12897, 2238}
The solution for the 9x9 board is unique.
See also https://oeis.org/A250000 Peaceable coexisting armies of queens:

Maximum No-Square Grid https://oeis.org/A227133
and A240443 (squares in any orientation)

Given a square grid with side n consisting of n^2 cells (or points), a(n) is the maximum number of points that can be painted so that no four of the painted ones form a square with sides parallel to the grid.
. X X X X X . X .
X . X . . X X X X
X X . . X . X . X
X . . X X X X . .
X X X . X . . X X
X . X X X . . . X
. X X . . X X . X
X X . X . X . X X
. X X X X X X X .
​
Squares in any orientation
x x . . x . x . x
. x . . x x x x .
x x x . . x . . x
x . x x x . . x x
. . . . x x . . .
. x . x x . . . x
x x x . x . . . x
x . x . . . . x x
x . . x x x x x .

Ramanujan’s Taxicab https://oeis.org/A046881
Smallest number that is sum of 2 positive distinct n-th powers in 2 different ways.

5 = 1^1 + 4^1 = 2^1 + 3^1;
65 = 1^2 + 8^2 = 4^2 + 7^2;
1729 = 1^3 + 12^3 = 9^3 + 10^3
https://oeis.org/A230563 Smallest number that is the sum of three positive n-th powers in at least two ways.
5 = 1^1 + 1^1 + 3^1 = 1^1 + 2^1 + 2^1.
27 = 1^2 + 1^2 + 5^2 = 3^2 + 3^2 + 3^2.
251 = 1^3 + 5^3 + 5^3 = 2^3 + 3^3 + 6^3.
2673 = 2^4 + 4^4 + 7^4 = 3^4 + 6^4 + 6^4.
1375298099 = 3^5 + 54^5 + 62^5 = 24^5 + 28^5 + 67^5.
160426514 = 3^6 + 19^6 + 22^6 = 10^6 + 15^6 + 23^6.

Smallest Cage Graph https://oeis.org/A000066

Smallest number of vertices in trivalent graph with girth (shortest cycle) = n.

Maximal number of moves required for the n X n generalization
of the sliding block 15-puzzle https://oeis.org/A087725

The 5×5 sliding block puzzle is unsolved. 152 <= a(5) <= 208

Maximum Determinant Problem https://oeis.org/A085000

Maximal determinant of an n X n matrix using the integers 1 to n^2.

Dedekind numbers https://oeis.org/A003182 Unsolved

number of simple games with n players in minimal winning form up to isomorphism.
​
For n=2, there are six monotonic Boolean functions and six antichains of subsets of the two-element set {x,y}
​
The function f(x,y) = false that ignores its input values and always returns false corresponds to the empty antichain Ø.
The logical conjunction f(x,y) = x ∧ y corresponds to the antichain { {x,y} } containing the single set {x,y}.
The function f(x,y) = x that ignores its second argument and returns the first argument corresponds to the antichain { {x} } containing the single set {x}
The function f(x,y) = y that ignores its first argument and returns the second argument corresponds to the antichain { {y} } containing the single set {y}
The logical disjunction f(x,y) = x ∨ y corresponds to the antichain { {x}, {y} } containing the two sets {x} and {y}.
The function f(x,y) = true that ignores its input values and always returns true corresponds to the antichain {Ø} containing only the empty set

Thanks for reading

For next year: What’s remarkable about 2026?
Let me recommend NJA Sloane’s paper “A Handbook of Integer Sequences” Fifty Years Later https://arxiv.org/abs/2301.03149

CITE THIS NOTEBOOK

Mathematical Games: 2025, other numbers and OEIS​
by Ed Pegg​
Wolfram Community, STAFF PICKS, January 31, 2025
​https://community.wolfram.com/groups/-/m/t/3370562