The following is some trivia but in fact, not so trivia, in this age of data science, data analytics, social media platforms, on-line gaming etc…If you decide to ponder over deep…you will become a giant mathematician or applied mathematician or of course, a computer science wunderkind..
The following is “picked out as it is” from a famous biography, (which regular readers of my blog will now know, perhaps, is a favorite mathematical biography for me)…A Beautiful Mind by Sylvia Nasar, biography of mathematical genius, John Forbes Nash, Jr, Nobel Laureate (Economics) and Abel Laureate:
“It was the great Hungarian-born polymath John von Neumann who first recognized that social behaviour could be analyzed as games. Von Neumann’s 1928 article on parlor games was the first successful attempt to derive logical and mathematical rules about rivalries. Just as William Blake saw the universe in a grain of sand, great scientists have often looked for clues in vast and complex problems in the small, familiar phenomena of daily life. Isaac Newton reached insights about the heavens by juggling wooden balls. Albert Einstein contemplated a boat paddling upriver. Von Neumann pondered the game of poker.
A seemingly trivial and playful pursuit like poker, von Neumann argued, might hold the key to more serious human affairs for two reasons. Both poker and economic competition require a certain type of reasoning, namely the rational calculation of advantage and disadvantage based on some internally consistent system of values (“more is better than less’). And, in both, the outcome for any individual actor depends not only on his own actions, but on the independent actions of others.
More than a century earlier, the French economist Antoine-Augustin Cournot had pointed out that problems of economic cnoice were greatly simplified when either none or a large number of other agents were present. Alone on his island, Robinson Crusoe does not have to worry about whose actions might affect him. Neither do Adam Smith’s butchers and bakers. They live in a world with so many others that their actions, in effect, cancel each other out. But when there is more than one agent but not so many that their influence may be safely ignored, strategic behavior raises a seemingly insoluble problem:”I think that he thinks that I think that he thingks,” and so forth…
So play games but think math ! 🙂
Nalin Pithwa
for the number you end up with.
which is contained between the co-ordinate axes, is bisected at the point of tangency.
for 
that are parallel to the line 
.
, where 
and 
, where 
may touch the curve 
.
and 
are lengths of the perpendiculars from origin on the tangent and normal to the curve 
respectively, prove that 
.
, 
is symmetrical about x-axis and has no real points for 
. If the tangent at the point t is inclined at an angle 
to OX, prove that 
. If the tangent at 
meets the curve again at Q, prove that the tangents at P and Q are at right angles.
other than 
on the curve 
meets the curve again at 
. The tangent at 
and so on. Show that the abscissae of 
form a GP. Also, find the ratio of area 
.
differ by less than 
.
, when 
, and 
when 
vanishes on an infinite set of points of the interval 
.
for 
