The search for these sequences in the times tables is motivated by well-known properties of Pascal's Triangle
, a simple triangular arrangement of numbers which finds important application in the fields of algebra, probability, financial arithmetic and calculus, to name just a few.
Topics encompass counting and proofs, sets and logic, graphs and functions, induction, algorithms with ciphers, binomial coefficients and Pascal's triangle
, counting techniques, recurrences, counting and geometry, trees, Euler's formula and applications, graph traversals, graph coloring, probability and expectation, and cardinality.
Szalay, Balancing in direction (1, -1) in Pascal's triangle
This is essentially the addition of all terms in the forty-second row of Pascal's Triangle
, or 2 [conjunction] 42 = 4398046511104.
Khayyam addressed the triangular array of binomial coefficients, which later became known as Pascal's triangle
and that stood at the base of probability.
Among the topics are counting and proofs, algorithms with ciphers, binomial coefficients and Pascal's triangle
, graph traversals, and probability ad expectation.
is an at of numbers in triangular form such that each number is the sum of the two above it.
Topics covered include the upward extension of Pascal's triangle
, a recurrence relation for powers of Fibonacci numbers, ways to make change for a million dollars, integer triangles, the probability of a perfect bridge hand, higher-dimensional tic-tac-toe, animal achievement and avoidance games, and an algorithm for solving Sudoku puzzles and polycube packing problems.
You do not need to be able to understand Pascal's Triangle
to appreciate what all that adds up to.
As a student in her school's Gifted and Talented program Dianna was introduced to the concepts of DNA Pascal's triangle
and the quantum enzyme code among other scientific phenomena.
We will take this to be a conjecture that the rows of Pascal's triangle
are infinitely log-concave, although we will later discuss the columns and other lines.
Proof without words: Refer to the following Pascal's Triangle
(2) and see the underlined numbers.