Math
536 Puzzles: Problem & Solution #1: Concerning a Check
A man went into a bank to cash a check. In handing over the money the cashier, by mistake, gave him dollars for cents and cents for dollars. He pocketed the money without examining it, and spent a nickel on his way home. He then found that he possessed exactly twice the amount of the check. He had no money in his pocket before going to the bank. What was the exact amount of that check?
- The first sheet uses a brute force method of stepping through all the values from 00.00 to 99.99 and looking for an integer solution
- The second sheet uses the relation below to drive towards a less brute force method
- The third sheet of the spread sheet shows how the solution is periodic like a set of gears. The constant term added is represented as the column offset number and is analogous to a number of teeth offset from zero position at the start of turning of the gears.
- The fourth sheet uses 5 and 7 to simplify the problem
Math
536 Puzzles: Problem & Solution #457: City Luncheons
Research Links
The clerks attached to the finn of Pilkins and Popinjay arranged that three of them would lunch together every day at a particular table so long as they could avoid the same three men sitting down twice together. The same number of clerks of Messrs. Radson, Robson, and Ross decided to do precisely the same, only with four men at a time instead of three. On working it out they found that Radson's staff could keep it up exactly three times as many
days as their neighbors. What is the least number of men there could have been in each staff?
Math
Langrange Multiplier Maximization Minimization Technique
Research Links
- Wikipedia: LaGrange Multiplier
- The Maximum Entropy Principle – The distribution with the maximum entropy is the distribution nature chooses
- Maximum Entropy Distribution for Random Variable of Extent [0,Infinity] and a Mean Value Mu
- Derivation of Nyquist 4KTBR Relation using Boltzmann 1/2KT Equipartition Theorem
- Derivation of the Normal Gaussian distribution from physical principles
Math
536 Puzzles: Problem & Solution #92: The Meeting Cars
Research Links
The Crackhams made their first stop at Bugleminster, where they were to spend the night at a friend's house. This friend was to leave home at the same time and ride to London to put up at the Crackhams' house. They took the same route, and each car went at its own uniform speed. They kept a look-out for one another, and met forty miles from Bugleminster. George that evening worked out the following little puzzle: "I find that if, on our respective arrivals, we had each at once proceeded on the return, journey at the same speeds we should meet at forty-eight miles from London. If this were so, what is the distance from London to Bugleminster?
Communication
Morse Code Encoding Tree reminds of Huffman Code
Morse code is made up of dots and dashes which can be thought of as 1's and 0's.
Morse Code is encoded along the lines of a Huffman code and thus one has to wonder if Huffman's paper was inspired by it.
