Fun Problems Research Articles

Fun Problems in Geometry and Beyond

We discuss fun problems, vaguely related to notions and theorems of a course in differential geometry. This paper can be regarded as a weekend "treasure chest" supplementing the course weekday lecture notes. The problems and solutions are not original, while their relation to the course might be so.

Call for Manuscripts: Problem of the Month

These nonroutine, fun problems invite multiple approaches and provide creative opportunities for individual students or math clubs. Send in your students' solutions–MT will celebrate their accomplishments by publishing selected work. Submit your student solutions directly to the Problem of the Month editors: Seán Madden, smadden@greeleyschool.org, and Ricardo Diaz, ricardo.diaz@unco.edu.

Digital dice: computational solutions to practical probability problems

Some probability problems are so difficult that they stump the smartest mathematicians. But even the hardest of these problems can often be solved with a computer and a Monte Carlo simulation, in which a random-number generator simulates a physical process, such as a million rolls of a pair of dice. This is what Digital Dice is all about: how to get numerical answers to difficult probability problems without having to solve complicated mathematical equations. Popular-math writer Paul Nahin challenges readers to solve twenty-one difficult but fun problems, from determining the odds of coin-flipping games to figuring out the behavior of elevators. Problems build from relatively easy (deciding whether a dishwasher who breaks most of the dishes at a restaurant during a given week is clumsy or just the victim of randomness) to the very difficult (tackling branching processes of the kind that had to be solved by Manhattan Project mathematician Stanislaw Ulam). In his characteristic style, Nahin brings the problems to life with interesting and odd historical anecdotes. Readers learn, for example, not just how to determine the optimal stopping point in any selection process but that astronomer Johannes Kepler selected his second wife by interviewing eleven women.The book shows readers how to write elementary computer codes using any common programming language, and provides solutions and line-by-line walk-throughs of a MATLAB code for each problem.Digital Dice will appeal to anyone who enjoys popular math or computer science. In a new preface, Nahin wittily addresses some of the responses he received to the first edition.

Sequencing

Sequencing

Development and evaluation of the problem identification picture cards method

In this paper the development and assessment of a new formative evaluation method called the problem identification picture cards (PIPC) method is described. This method enables young children to express both usability and fun problems while playing a computer game. The method combines the traditional thinking-aloud method with picture cards that children can place in a box to indicate that there is a certain type of problem. An experiment to assess this method shows that children may express more problems (verbally, or with a picture card, or with a combination of a picture card and a verbalisation) with the PIPC method than without this method (in which they can only indicate problems verbally). Children in the experiment did not just replace verbalisations by using the provided picture cards and some children preferred to use the PIPC method during the test instead of the standard thinking-aloud method. The PIPC method or some aspects of the method could be a good instrument to increase the amount of information expressed by young children during an evaluation.

Developing a coding scheme for detecting usability and fun problems in computer games for young children

This article describes the development and assessment of a coding scheme for finding both usability and fun problems through observations of young children playing computer games during user tests. The proposed coding scheme is based on an existing list of breakdown indication types of the detailed video analysis method (DEVAN). This method was developed to detect usability problems in task-based products for adults. However, the new coding scheme for children's computer games takes into account that in games, fun, in addition to usability, is an important factor and that children behave differently from adults. Therefore, the proposed coding scheme uses 8 of the 14 original breakdown indications and has 7 new indications. The article first discusses the development of the new coding scheme. Subsequently, the article describes the reliability assessment of the coding scheme. The any-two agreement measure of 38.5% shows that thresholds for when certain user behavior is worth coding will be different for different evaluators. However, the any-two agreement of .92 for a fixed list of observation points shows that the distinction between the available codes is clear to most evaluators. Finally, a pilot study shows that training can increase any-two agreement considerably by decreasing the number of unique observations, in comparison with the number of agreed upon observations.

Identifying usability and fun problems in a computer game during first use and after some practice

This paper describes an experiment to discover the change in the types of detected problems and the attitude of children towards a game when user testing a computer game for young children during first use and after they have practiced with a game. Both the numbers of different types of identified problems and the severity of the problems are investigated. Based on this knowledge, practitioners could adapt the set up of their user tests to effectively find as many aspects of the game as possible that merit change, according to the aims of the developers. The study shows that usability problems caused by a lack of knowledge were more often identified during first use. Furthermore, fun problems related to a too-high challenge level may disappear after some practice, whereas fun problems caused by the game taking over control for too long while the user wants to proceed playing the game were identified more often after some practice. The study shows that the impact severity of problems detected during first use was higher than when children had more practice with a game. As a result of these changes in experienced problems the commonly used measures efficiency, effectiveness and satisfaction increased when children had practiced with the game. Finally, the study also shows that the set of most severe problems identified during first use may be radically different from the set of most severe problems identified after some practice.

Truck Drivers, a Straw, and Two Glasses of Water

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsKevin IgaKevin Iga (kiga@pepperdine.edu) received his Ph.D. from Stanford University in 1998, studying four-dimensional topology under Ralph Cohen. Since then, he has taught at Pepperdine University (Malibu, CA 90263). His current research emphases include differential topology and the mathematics behind a physics theory called supersymmetry, but basically he works on any fun problem that comes his way.Kendra KillpatrickKendra Killpatrick (Kendra.Killpatrick@pepperdine.edu) received her Ph.D. from the University of Minnesota in 1998 in combinatorics. She spent three years after graduation as a post-doctoral student at Colorado State University and then began teaching at Pepperdine in 2002. Her main areas of research are enumerative combinatorics, especially permutation statistics and certain differential posets called Fibonacci posets. Outside mathematics, she loves to run marathons!

Figures of Constant Width on a Chessboard

Every row and column of the board has only two occupied squares. But if we also consider the diagonals with slope +1 or -1, we see that each of them contains either zero or two squares of the figure. We say that Figure 1 has constant width three by rows and columns, and that Figure 2 has constant width two by rows, columns, and diagonals. The first figure is easy to generalize in the sense that the figure formed by all the squares of a w x w chessboard has constant width w by rows and columns. On the other hand, if we also consider the two diagonal directions as in Figure 2, the problem of finding figures of constant width higher than two becomes considerably harder. Is it possible, for example, to find a nonempty figure on some n x n chessboard such that every row, column, and diagonal intersects it in zero or three squares? This is a fun problem to think about. We encourage the reader to try it before he or she continues reading. As we will show here, the answer turns out to be yes, even for widths higher than three. The impatient reader might want to take a look at Figures 6, 11, 12, and 13. In order to state our main result in a precise form, we introduce some terminology. We designate as a figure any set of squares in an n x n chessboard. A figure F has constant width w if every row, column, or diagonal intersects it in 0 or w squares. To be more exact, F is of type (n, k, w) if it has constant width w in a chessboard of size n x n and has kw squares. Observe that k is also the number of nonempty rows (or columns) in the chessboard.

Essays on the Motion of Celestial Bodies

Essays on the Motion of Celestial Bodies

Dynamic programming for pennies a day

An elegant solution to a fun problem is presented as a way to introduce dynamic programming. This problem, the Nine Tails, can be used as an introductory supplement to the traditional examples offered in many textbooks which cover dynamic programming.

A simple proof for a fun problem

Pick any four one-digit numbers (just so no three digits are the same) and write them down as one four-digit number. Even better, have your fifth- or sixth-grade pupiJs choose the numbers. Now take this number and rearrange the four digits so that they are ordered high to low as you read from left to right. Next, off to the side, again rearrange the digits, this time ordering low to high, and then drop the highest digit so that you have a three-digit number. Now go back and subtract this three-digit number from the high-to-lowordered four-digit number. For example, if you had chosen 3781 your subtraction problem would be 8731 − 137 =?.

Book ReviewFundamental Problems in Scanning. Compiled and Edited by GottschalkA. and BeckR. N., pp. xvi + 410, illus., 1968 (Springfield, Illinois, Charles C. Thomas), $26.75.

I wish it had been possible to go to the meeting at the Argonne Cancer Research Hospital in 1965, of which this magnificent book is a record. I first came across the book referred to in a shorthand fashion as “Fun Problems”—certainly the meeting must have been fun as well as fundamental. Although it is primarily a collection of presented papers, they are invited papers by authorities on their subjects and arranged with linking papers (prepared afterwards to bring it more up-to-date than one would expect from the date of the meeting) to present a logical review of radioisotope scanning with particular reference to the limitations of the method and potential areas of improvement. The emphasis throughout is on the fundamental problems in the field, actual systems and examples being used primarily to highlight these problems rather than as an end in themselves.