Threes!, Fives, 1024!, and 2048 are Hard

We analyze the computational complexity of the popular computer games Threes!, 1024!, 2048 and many of their variants. For most known versions expanded to an m x n board, we show that it is NP-hard to decide whether a given starting position can be played to reach a specific (constant) tile value.Comment: 14 pages, 9 figure

2048 Without New Tiles Is Still Hard

We study the computational complexity of a variant of the popular 2048 game in which no new tiles are generated after each move. As usual, instances are defined on rectangular boards of arbitrary size. We consider the natural decision problems of achieving a given constant tile value, score or number of moves. We also consider approximating the maximum achievable value for these three objectives. We prove all these problems are NP-hard by a reduction from 3SAT. Furthermore, we consider potential extensions of these results to a similar variant of the Threes! game. To this end, we report on a peculiar motion pattern, that is not possible in 2048, which we found much harder to control by similar board designs

Making Change in 2048

The 2048 game involves tiles labeled with powers of two that can be merged to form bigger powers of two; variants of the same puzzle involve similar merges of other tile values. We analyze the maximum score achievable in these games by proving a min-max theorem equating this maximum score (in an abstract generalized variation of 2048 that allows all the moves of the original game) with the minimum value that causes a greedy change-making algorithm to use a given number of coins. A widely-followed strategy in 2048 maintains tiles that represent the move number in binary notation, and a similar strategy in the Fibonacci number variant of the game (987) maintains the Zeckendorf representation of the move number as a sum of the fewest possible Fibonacci numbers; our analysis shows that the ability to follow these strategies is intimately connected with the fact that greedy change-making is optimal for binary and Fibonacci coinage. For variants of 2048 using tile values for which greedy change-making is suboptimal, it is the greedy strategy, not the optimal representation as sums of tile values, that controls the length of the game. In particular, the game will always terminate whenever the sequence of allowable tile values has arbitrarily large gaps between consecutive values

Tracks from hell - when finding a proof may be easier than checking it

We consider the popular smartphone game Trainyard: a puzzle game that requires the player to lay down tracks in order to route colored trains from departure stations to suitable arrival stations. While it is already known [Almanza et al., FUN 2016] that the problem of finding a solution to a given Trainyard instance (i.e., game level) is NP-hard, determining the computational complexity of checking whether a candidate solution (i.e., a track layout) solves the level was left as an open problem. In this paper we prove that this verification problem is PSPACE-complete, thus implying that Trainyard players might not only have a hard time finding solutions to a given level, but they might even be unable to efficiently recognize them

On the Complexity of Two Dots for Narrow Boards and Few Colors

Two Dots is a popular single-player puzzle video game for iOS and Android. A level of this game consists of a grid of colored dots. The player connects two or more adjacent dots, removing them from the grid and causing the remaining dots to fall, as if influenced by gravity. One special move, which is frequently a game-changer, consists of connecting a cycle of dots: this removes all the dots of the given color from the grid. The goal is to remove a certain number of dots of each color using a limited number of moves. The computational complexity of Two Dots has already been addressed in [Misra, FUN 2016], where it has been shown that the general version of the problem is NP-complete. Unfortunately, the known reductions produce Two Dots levels having both a large number of colors and many columns. This does not completely match the spirit of the game, where, on the one hand, only few colors are allowed, and on the other hand, the grid of the game has only a constant number of columns. In this paper, we partially fill this gap by assessing the computational complexity of Two Dots instances having a small number of colors or columns. More precisely, we show that Two Dots is hard even for instances involving only 3 colors or 2 columns. As a contrast, we also prove that the problem can be solved in polynomial-time on single-column instances with a constant number of goals

Threes! Fives, 1024! and 2048 are Hard

We analyze the computational complexity of the popular computer games Threes! 1024! 2048 and many of their variants. For most known versions expanded to an m×n board, we show that it is NP-hard to decide whether a given starting position can be played to reach a specific (constant) tile value.SCOPUS: ar.jinfo:eu-repo/semantics/publishe

The oral language productive vocabulary profile of children starting school : a resource for teachers

The oral language proficiency of students in early education is crucial as teachers draw on this as a resource when developing literacy. There is a need to better understand what this oral language resource consists of at school entry, particularly the diversity amongst children so as to address inequalities. This article reports a study on a key component of oral language, namely productive vocabulary. It profiles the oral language vocabulary in approximately 3.6 million words produced by a large sample of almost 800 children under the age of five. The results are reported in a productive vocabulary resource, structured as a list of 2767 vocabulary targets. This profile represents highly productive vocabulary presumably known by most children as well as more advanced vocabulary not part of every child's oral language. The article demonstrates the pedagogical implications of this research in the context of the National Literacy Learning Progressions of the Australian National Curriculum

Assessing the potential of community mobilisation with women's groups to improve child growth among underserved tribal communities of Eastern India

Background: India is home to one-third of the world’s undernourished children. Rural tribal areas are disproportionately affected. Community-based behaviour change interventions are central to addressing undernutrition. Most interventions have used didactic educational methods but have had a limited impact; fewer studies have tested participatory approaches. This thesis explores the potential of a participatory intervention to reduce child undernutrition in rural tribal communities of Eastern India. Methods: We conducted a cross-sectional nutrition survey of 36 village-clusters in three districts of Jharkhand and Orissa: 18 clusters had been exposed to community mobilisation with women’s groups to improve child health and nutrition; 18 control clusters matched the intervention areas on population and health-service characteristics. We also conducted focus groups with caregivers of young children. Results: There were no group differences for child anthropometry. Levels of undernutrition were extremely high: 40% of children were experiencing global acute malnutrition, 60% were stunted, and 24% had mid-arm-circumference measurements in the moderate-severe malnutrition category. There were significant group differences for hand washing, water treatment, birth spacing, measles vaccination and awareness of child undernutrition that favoured the intervention group; there were no differences for child feeding practices, health-service uptake or child morbidity. The analyses identified a multitude of undernutrition determinants including strong protective effects of hand washing, and diarrhoea as a major risk factor. The focus groups revealed extreme food insecurity, problematic feeding and hygiene practices, and inadequate health services. Conclusion: Community mobilisation with women’s groups does not appear to have reduced child undernutrition in this context, but has the potential to improve important nutrition behaviours. There is scope to improve and combine this intervention with complementary strategies, but until the wider problems of food insecurity, poverty and poor health-services are addressed community mobilisation with women’s groups, on its own, is unlikely to meaningfully impact on undernutrition

The historia plantarum generalis of John Ray :Book I - a translation and commentary

PhD ThesisAfter a preface explaining the origins of my interest in John Ray, and the general principles on which I propose to proceed, the thesis consists, firstly, of a brief account of Ray's life and work, with particular attention to the state of botanical studies in his day, and of his main contributions to them. The main part of the thesis (Volumes 1 and 2) then takes the form of a commentary on Book I of the Historia Plantarum Generalis and the three subsequent tables interpolated into Volume I at the suggestion of Dr. Tancred Robinson; the third volume of the thesis consists of my translation and of a photocopy of the text. Since Ray lived before Linnus, to whom we owe our modern binomial system of nomenclature, and because of the confused state of botanical nomenclature up to Ray's time, much of the commentary consists of the identification of the plants mentioned by Ray as examples of various botanical and horticultural processes. However, I also discuss the accuracy of Ray's observations and explanations of the various processes in the light of modern scientific views, and assess their place in the development of botanical science. Ray's sources and his use of them would make an interesting thesis in itself. Since, however, to comment upon them all in detail would have made an already lengthy thesis even lengthier, I have singled out for detailed analysis the material cited by Ray from his contemporary Malpighi and the first century A.D. Roman writer, Pliny the Elder. Brief biographies of all Ray's sources, both ancient and modern, are also given. Finally, I have included diagrams where I felt this would help to clarify what Ray says

Reports of explorations and surveys, to ascertain the most practicable and economical route for a railroad from the Mississippi River to the Pacific Ocean. Made under the direction of the Secretary of War, in 1853-4, according to Acts of Congress of March 3, 1853, May 31, 1854, and August 5, 1854

Explorations of Railroad Routes from the Mississippi to the Pacific. [791-801] Journals of expeditions, including descriptions of Indians; a northern route from Minnesota, along the Upper Missouri, to Washington; a central route from Missouri, across the Great Basin, to California; a southern route from the Red River, along the Rio Grande and Colorado Rivers, to southern California; a report on Indian tribes of the Southwest is included in pt. 3 (Serials 760 and 793)