We formulate a Sudoku-puzzle-solving algorithm that implements a hierarchy of four simple logical rules commonly used by humans. The difficulty of a puzzle is determined by recording the sophistication and relative frequency of the methods required to solve it. Four difficulty levels are established for a puzzle, each pertaining to a range of numerical values returned by the solving function.
Like humans, the program begins solving each puzzle with the lowest level of logic necessary. When all lower methods have been exhausted, the next echelon of logic is implemented. After each step, the program returns to the lowest level of logic. The procedure loops until either the puzzle is completely solved or the techniques of the program are insufficient to make further progress.
The construction of a Sudoku puzzle begins with the generation of a so-lution by means of a random-number-based function. Working backwards from the solution, numbers are removed one by one, at random, until one of several conditions, such as a minimum difficulty rating and a minimum number of empty squares, has been met. Following each change in the grid, the difficulty is evaluated. If the program cannot solve the current puzzle, then either there is not a unique solution, or the solution is beyond the grasp of the methods of the solver. In either case, the last solvable puzzle is restored and the process continues.
Uniqueness is guaranteed because the algorithm never guesses. If there
is not sufficient information to draw further conclusions—for example, an arbitrary choice must be made (which must invariably occur for a puzzle with multiple solutions)—the solver simply stops. For obvious reasons, puzzles lacking a unique solution are undesirable. Since the logical tech-niques of the program enable it to solve most commercial puzzles (for ex-ample, most “evil” puzzles from Greenspan and Lee [2008]), we assume that demand for puzzles requiring logic beyond the current grasp of the solver is low. Therefore, there is no need to distinguish between puzzles requiring very advanced logic and those lacking unique solutions.
The text of this paper appears on pp. 381–394.
Pp. 237–248 can be found on the Tools for Teaching 2008 CD-ROM.
The Impending Effects of North Polar Ice Cap Melt
Benjamin Coate Nelson Gross Megan Longo
College of Idaho Caldwell, ID
Advisor: Michael P. Hitchman
Abstract
Because of rising global temperatures, the study of North Polar ice melt has become increasingly important.
• How will the rise in global temperatures affect the melting polar ice caps and the level of the world’s oceans?
• Given the resulting increase in sea level, what problems should metropoli-tan areas in a region such as Florida expect in the next 50 years?
We develop a model to answer these questions.
Sea level will not be affected by melting of the floating sea ice that makes up most of the North Polar ice cap, but it will be significantly affected by the melting of freshwater land ice found primarily on Greenland, Canada, and Alaska. Our model begins with the current depletion rate of this freshwater land ice and takes into account
• the exponential increase in melting rate due to rising global temperatures,
• the relative land/ocean ratios of the Northern and Southern Hemispheres,
• the percentage of freshwater land ice melt that stays in the Northern Hemisphere due to ocean currents, and
• thermal expansions of the ocean due to increased temperatures on the top layer.
The UMAP Journal 29 (3) (2008) 237–247. c!Copyright 2008 by COMAP, Inc. All rights reserved.
We construct best- and worst-case scenarios. We find that in the next 50 years, the relative sea level will rise 12 cm to 36 cm.
To illustrate the consequences of such a rise, we consider four Florida coastal cities: Key West, Miami, Daytona Beach, and Tampa. The problems that will arise in many areas are
• the loss of shoreline property,
• a rise of the water table,
• instability of structures,
• overflowing sewers,
• increased flooding in times of tropical storms, and
• drainage problems.
Key West and Miami are the most susceptible to all of these effects. While Daytona Beach and Tampa are relatively safe from catastrophic events, they will still experience several of these problems to a lesser degree.
The effects of the impending rise in sea level are potentially devastating;
however, there are steps and precautions to take to prevent and minimize destruction. We suggest several ways for Florida to combat the effects of rising sea levels: public awareness, new construction codes, and prepared-ness for natural disasters.
Introduction
We consider for the next 50 years the effects on the Florida coast of melting of the North Polar ice cap, with particular attention to the cities noted. This question can be broken down into two more-detailed questions:
• What is the melting rate, and its effects on sea level?
• How will the rising water affect the Florida cities, and what can they do to counteract and prepare?
Our models use the geophysical data in Table 1 and the elevations of cities in Table 2.
Table 1.
Geophysical data.
Entity Value Unit
Total volume of ice caps 2.422 × 107 km3 Surface area of world’s oceans 3.611 × 108 km2 Surface area of ice on Greenland 1.756 × 106 km2 Volume of ice on Greenland 2.624 × 106 km3
Elevations of Florida cities.
City Average Maximum
elevation (m) elevation (m)
Key West 2.44 5.49
Miami 2.13 12.19
Daytona Beach 2.74 10.36