It reminds of the “unreasonable ineffectiveness of mathematics” as explained by Eugene Wigner long back ! This in itself is a strong motivation to learn math!!
Six boxes are numbered 1 through 6. How many ways are there to put 20 identical balls into these boxes so that none of them is empty?
Problem 2:
How many ways are there to distribute n identical balls in m numbered boxes so that none of the boxes is empty?
Problem 3:
Six boxes are numbered 1 through 6. How many ways are there to distribute 20 identical balls between the boxes (this time some of the boxes can be empty)?
I have a bright kid working towards his IITJEE Foundation math. Some days back I had suggested to him to solve some hard word problems based on simultaneous equations from a very old classic text. He started on his own, almost with some guidance from me. Until he attacked very well — those “age” kind of problems. Father’s age vs. son’s age, etc. But, in this case, he suddenly sprang to his feet: He almost yelled, “Sir! Please check my solution! I am getting the answer as husband’s age is 48 and wife’s age is 23!” Actually, I too was a bit shocked; but, I checked his calculations in detail; they were mathematically correct. It suddenly flashed in my head:
“Do you know Vedant? Mathematics is not human! It has no emotions, no feelings, whatsoever! I told him the following quote of Bertrand Russell: ” Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty, cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.””
And, of course, equations//numbers don’t lie! 🙂 🙂 🙂
It is said that one of the best sources of inspiration about mathematics and “men” of mathematics is the book, “Menn of Mathematics” by E. T. Bell. (John Nash, Jr., Nobel laureate, Abel Laureate, genius mathematician, who passed away some time back, had also read this classic as a young boy. You might have known him through the movie, A Beautiful Mind starring Russell Crowe). But, there have also been “really rare” “women of mathematics” like Emmy Noether (who had passed away unsung, unheard; she also passed away due to some cancer), and just recently, eminent mathematician Maryam Mirzakhani, of Iranian origin, Fields Medallist, and professor at Stanford University, has also passed away at a young age of 40 due to cancer. It is said that mathematicians are loners; but she proved it otherwise: she was a professor, a wife, a mother also…
I wish someone writes a classic called “Women of Mathematics” also just like long back E. T. Bell wrote the classic, “Men of Mathematics”.
Below is an obit on Maryam Mirzakhani in The Times of India online:
Pre-RMO days are back again. Here is a list of some of my random thoughts:
Problem 1:
There are five different teacups, three saucers, and four teaspoons in the “Tea Party” store. How many ways are there to buy two items with different names?
Problem 2:
We call a natural number “odd-looking” if all of its digits are odd. How many four-digit odd-looking numbers are there?
Problem 3:
We toss a coin three times. How many different sequences of heads and tails can we obtain?
Problem 4:
Each box in a 2 x 2 table can be coloured black or white. How many different colourings of the table are there?
Problem 5:
How many ways are there to fill in a Special Sport Lotto card? In this lotto, you must predict the results of 13 hockey games, indicating either a victory for one of two teams, or a draw.
Problem 6:
The Hermetian alphabet consists of only three letters: A, B and C. A word in this language is an arbitrary sequence of no more than four letters. How many words does the Hermetian language contain?
Problem 7:
A captain and a deputy captain must be elected in a soccer team with 11 players. How many ways are there to do this?
Problem 8:
How many ways are there to sew one three-coloured flag with three horizontal strips of equal height if we have pieces of fabric of six colours? We can distinguish the top of the flag from the bottom.
Problem 9:
How many ways are there to put one white and one black rook on a chessboard so that they do not attack each other?
Problem 10:
How many ways are there to put one white and one black king on a chessboard so that they do not attack each other?
when 
![\sqrt[4]{97-56\sqrt{3}}](https://s0.wp.com/latex.php?latex=%5Csqrt%5B4%5D%7B97-56%5Csqrt%7B3%7D%7D&bg=ffffff&fg=000&s=0 "\sqrt[4]{97-56\sqrt{3}}")
, prove that 
.