Pulpit Commentaries Verses Now He begins to reveal unto Daniel these things are going to transpire, as far as the future is concerned.

It turns out to be quite easy about one page of code for the main idea and two pages for embellishments using two ideas: Sudoku Notation and Preliminary Notions First we have to agree on some notation.

A Sudoku puzzle is a grid of 81 squares; the majority of enthusiasts label the columnsthe rows A-I, and call a collection of nine squares column, row, or box a unit and the squares that share a unit the peers.

A puzzle leaves some squares blank and fills others with digits, and the whole idea is: A puzzle is solved if the squares in each unit are filled with a permutation of the digits 1 to 9. That is, no digit can appear twice in a unit, and every digit must appear once.

This implies that each square must have a different value from any of its peers. Here are the names of the squares, a typical puzzle, and the solution to the puzzle: For example, here are the units and peers for the square C2: So read this assignment statement as "units is a dictionary where each square maps to the list of units that contain the square".

Similarly, read the next assignment statement as "peers is a dictionary where each square s maps to the set of squares formed by the union of the squares in the units of s, but not s itself". It can't hurt to throw in some tests they all pass: Actually we need two representations: First, a textual format used to specify the initial state of a puzzle; we will reserve the name grid for this.

Second, an internal representation of any state of a puzzle, partially solved or complete; this we will call a values collection because it will give all the remaining possible values for each square.

For the textual format grid we'll allow a string of characters with indicating a digit, and a 0 or period specifying an empty square.

All other characters are ignored including spaces, newlines, dashes, and bars. So each of the following three grid strings represent the same puzzle: One might think that a 9 x 9 array would be the obvious data structure. But squares have names like 'A1', not 0,0. Therefore, values will be a dict with squares as keys.

The value of each key will be the possible digits for that square: This collection of digits could be represented by a Python set or list, but I chose instead to use a string of digits we'll see why later.

Here is the code to parse a grid into a values dict: Those with experience solving Sudoku puzzles know that there are two important strategies that we can use to make progress towards filling in all the squares: These updates to A2 may in turn cause further updates to its peers, and the peers of those peers, and so on.

This process is called constraint propagation. The function assign values, s, d will return the updated values including the updates from constraint propagationbut if there is a contradiction--if the assignment cannot be made consistently--then assign returns False.

For example, if a grid starts with the digits ' It turns out that the fundamental operation is not assigning a value, but rather eliminating one of the possible values for a square, which we implement with eliminate values, s, d. Once we have eliminate, then assign values, s, d can be defined as "eliminate all the values from s except d".

Return values, except return False if a contradiction is detected. I picked the first example from a list of easy puzzles from the fine Project Euler Sudoku problem and tried it: Unfortunately, that will not always be the case.

Here is the first example from a list of hard puzzles: We could try to code more sophisticated strategies. For example, the naked twins strategy looks for two squares in the same unit that both have the same two possible digits.

Coding up strategies like this is a possible route, but would require hundreds of lines of code there are dozens of these strategiesand we'd never be sure if we could solve every puzzle.

Search The other route is to search for a solution:A set of four complete Maths 11+ exam papers with answers.

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0 items. The 11+ examination is used to select students for Year 7 entry into schools across the country. We 5/5(31). Professional chemists will be m ore precise and say that fluorine is the most “electronegative” element, meaning that it is the most aggressive electron “thief,” because the alkali metals are the most “electropositive” elements, meaning their affinity to “give away” an electron, and electronegativity and electropositivity are not easy to compare, as far as “reactivity” goes.

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