Progressive Randomization of a Deck of Playing Cards: Experimental Tests and Statistical Analysis of the Riffle Shuffle

The question of how many shuffles are required to randomize an initially ordered deck of cards is a problem that has fascinated mathematicians, scientists, and the general public. The two principal theoretical approaches to the problem, which differed in how each defined randomness, has led to statistically different threshold numbers of shuffles. This paper reports a comprehensive experimental analysis of the card randomization problem for the purposes of determining 1) which of the two theoretical approaches made the more accurate prediction, 2) whether different statistical tests yield different threshold numbers of randomizing shuffles, and 3) whether manual or mechanical shuffling randomizes a deck more effectively for a given number of shuffles. Permutations of 52-card decks, each subjected to sets of 19 successive riffle shuffles executed manually and by an auto-shuffling device were recorded sequentially and analyzed in respect to 1) the theory of runs, 2) rank ordering, 3) serial correlation, 4) theory of rising sequences, and 5) entropy and information theory. Among the outcomes, it was found that: 1) different statistical tests were sensitive to different patterns indicative of residual order;2) as a consequence, the threshold number of randomizing shuffles could vary widely among tests;3) in general, manual shuffling randomized a deck better than mechanical shuffling for a given number of shuffles;and 4) the mean number of rising sequences as a function of number of manual shuffles matched very closely the theoretical predictions based on the Gilbert-Shannon-Reed (GSR) model of riffle shuffles, whereas mechanical shuffling resulted in significantly fewer rising sequences than predicted.

Uncovering and Displaying the Coherent Groups of Rank Data by Exploratory Riffle Shuffling

Let n respondents rank order d items, and suppose that . Our main task is to uncover and display the structure of the observed rank data by an exploratory riffle shuffling procedure which sequentially decomposes the n voters into a finite number of coherent groups plus a noisy group: where the noisy group represents the outlier voters and each coherent group is composed of a finite number of coherent clusters. We consider exploratory riffle shuffling of a set of items to be equivalent to optimal two blocks seriation of the items with crossing of some scores between the two blocks. A riffle shuffled coherent cluster of voters within its coherent group is essentially characterized by the following facts: 1) Voters have identical first TCA factor score, where TCA designates taxicab correspondence analysis, an L1 variant of correspondence analysis;2) Any preference is easily interpreted as riffle shuffling of its items;3) The nature of different riffle shuffling of items can be seen in the structure of the contingency table of the first-order marginals constructed from the Borda scorings of the voters;4) The first TCA factor scores of the items of a coherent cluster are interpreted as Borda scale of the items. We also introduce a crossing index, which measures the extent of crossing of scores of voters between the two blocks seriation of the items. The novel approach is explained on the benchmarking SUSHI data set, where we show that this data set has a very simple structure, which can also be communicated in a tabular form.

Short-Term Disturbance Effects of Outdoor Education Stream Classes on Aquatic Macroinvertebrates

Outdoor education programs frequently offer stream classes that teach students how to assess water quality based on the composition of aquatic macroinvertebrate communities. Repeatedly using the same site for stream classes can cause disturbance that could negatively impact aquatic macroinvertebrates. However, the impact of outdoor education stream classes on short term temporal trends of aquatic macroinvertebrates has not been evaluated. Our objective was to quantify whether outdoor education stream classes caused localized and short-term impacts on the aquatic macroinvertebrate community. We sampled aquatic macroinvertebrates over a five day period in May 2014 from an impacted riffle subjected to repeated substrate disturbance by outdoor education stream classes and an unimpacted riffle that was not subjected to stream classes within an agricultural stream in central Ohio. We did not observe a difference in macroinvertebrate community response variables between different time periods or among days within the impacted riffle as part of our within riffle analysis. We documented that macroinvertebrate abundance and dorsoventrally flattened clinger abundance was greater within the unimpacted riffle than the impacted riffle. Macroinvertebrate evenness was greater in the impacted riffle than the unimpacted riffle. In the impacted riffle percent clingers was greater on Monday than on Friday. Conversely, in the unimpacted riffle percent clingers was greater on Friday than on Monday. Our results indicated that outdoor education stream classes did not alter short term temporal trends of macroinvertebrate community structure within an impacted riffle, but the stream classes resulted in differences in community structure between an impacted and unimpacted riffle. Our results suggest that outdoor education centers should avoid repeatedly using the same site for their stream classes to prevent negatively impacting aquatic macroinvertebrates.

Study on the Utilization of Low Grade Tin Ore Results Washing(SHP)with a Shaking Table to Improve Recovery and Grade of Tin Ore:Case Study in Toboali Washing Plant PT Timah Tbk Bangka

Shaking table is a technology that is still used in processing of tin ore using water as a medium.In the application at PT Timah Tbk,the shaking table is still used to process low grade tin ore into tin ore with a standard smelting grade of PT Timah(72%-74%Sn).In processing tin ore using a shaking table,valuable minerals and impurity minerals are separated based on differences in specific gravity,size and shape of the mineral.To get the recovery of valuable minerals(recovery)and optimal grades,it is strongly influenced by the separation variables of tin ore minerals.These variables include riffle,slope of deck,washing water rate,stroke length,and deck movement speed.This study aims to vary the number of riffles,table slope and time in a laboratory scale to obtain optimal recovery and grade of tin ore.Tin ore samples were obtained from the washing residue of the Toboali Washing Plant tin ore.In the experiment,the numbers of riffles set were 16(R1),24(R2)and 34(R3).The table slopes were 2°(S1),3°(S2),and 4°(S3).Time variations were 15(T1),20(T2)and 25 min(T3).The relationship between the number of riffles and the grade can be determined through quadratic polynomial regression analysis with the functionŶ=-20.60+2.26X+(-0.04X2),while the relationship between the number of riffles and recovery is determined by the functionŶ=-173.33+17.376 X+(-0.315X2).Optimal levels are obtained at variations in the number of riffle 26 by 66.43%,with levels of 8.45%Sn.Optimal recovery is obtained on the variation of the number of riffle 28 with a recovery of 66.43%.