The 19th century saw a change in mathematics in a way that it was more organized and
Various tricks and methods were introduced so as to make things easy to understand this
subject of great mystery.
Since the people then were educated from a young age and had access to almost all the
major discoveries and because the world had long ago understood the true importance of
mathematics as a subject, the era from the 19th century dedicated their lives to make the
data more organized and the mathematical operations more fun to understand.
Two of the well known concepts were the Venn diagram and the Tower of Hanoi
Venn Diagram
In 1880, British Mathematician John Venn introduced the idea of Venn diagram in his paper,
“On the Diagrammatic and Mechanical Representation of Propositions and Reasonings”
The venn diagram is a way of showing the relation between different sets of things through
overlapping circles.
The concept basically aims to visually denote the similarities and differences between sets.
The circles that overlap have a common property while the part of the circles that don’t
overlap do not share those traits.
The Venn diagram allows the data to be represented in an organized and simple manner.
Two circle Venn Diagrams can represent categorical positions such as, ‘All A are B’, ‘Some
A are B’ and ‘No A are B’
John Venn’s interpretation on the Venn diagram was then further taken up by later
The Tower of Hanoi
The Tower of Hanoi is a game invented by a French mathematician named Édouard Lucas.
The Tower of Hanoi is a mathematical game or puzzle consisting of three rods and a number
of disks of various diameters, which can slide onto any rod. The challenger is presented
with, let’s say three disks arranged in one of the poles, in order of the size such that the
largest disk is on the bottom.
The three disks must be moved one disk at a time so as to recreate the starting arrangement
using the smallest possible number of moves. In addition to this, the players can only place
a disk on top of a larger disk or onto an empty pole.
Solving the puzzles
The Tower of Hanoi can be solved using the formula 2^n-1 (where n is equal to the number
of moves), which will give the minimum number of moves.
Hence taking 3 disks, the puzzle can be solved in only 7 moves.
According to a popular legend, if the monks of a temple succeed in moving 64 disks from
one pole to another, in line with the rules, the world will end. However even using the best
strategy and moving one disk per second, they would take 585 billion years to complete the
In 1963, American Mathematician, David W Henderson outlined the connection between
symmetrical Venn diagrams and prime numbers, and then in 2003 Jerrold Griggs, Charles
Killian and Carla Savage showed that symmetrical venn diagrams exist for all planes.
As for the Tower of Hanoi, in 1959 Erik Frank Russell published his story “Now Inhale” which
is a short story about an alien allowed to play a short version of the puzzle before his
In an episode of BBC’s Doctor Who, the villain, Celestial Toymaker, forces the Doctor to play
a ten disk version of this game.
As it can be noticed, the mathematicians now accessed with the knowledgeable discoveries
of their ancestors, aimed to make the subject more comprehensible.
Through different experiments and practices they arrived at conclusions that are now widely
used throughout the world.