In this paper we give a complete solution to the conjecture of Evans made in 1960 that if n-1 cells of an nxn matrix are preassigned with no element repeated in any row or column then the remaining n 2 -n+ 1 cells can be filled so as to produce a latin square. We in fact prove the stronger statement that n cells can be preassigned except in certain cases which we specify.

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We prove that a partial Steiner triple system S of ordern can be embedded in a Steiner triple system T of any given admissible order greater than 4n. Furthermore, if G(S), the missing‐edge graph of S, has the property that Δ(G)<[12(n+1)]and|E(G)|⩽([12(n‐1)]2), then S can be embedded in a Steiner triple system of order 2n + 1, provided that 2n + 1 is admissible. We also prove that if there is a partial Steiner triple system of order n with v triples then there is an equitable partial Steiner triple system of order n with v triples. This result, interesting in itself, is used in the proof of the above theorems.

Virtual reality (VR)-based rehabilitation is a growing technological field, which gradually becomes integrated into existing programs. However, technology has to support human behavior and -needs, including social relatedness, to achieve health-related outcomes. Elderly people have high risk of loneliness, and VR has technological affinity for natural social interaction. Previous studies have relied on competitiveness rather than collaborative elements, but research shows that competitiveness can lead to (feelings of) stress and aggressive behavior in some individuals. This article presents a mixed methods study to gather end-user feedback on a social VR scenario that encourages inter-player collaboration on a virtual tandem bike. Outpatients ($$n=11$$ n = 11 , 64% males, $$60\pm 11$$ 60 ± 11 years) were invited to participate with a co-player (friend or family). Participants biked on average 10.7 (± 3) minutes with a mean speed of 14.8 kmph (± 5.8). The results indicate potential and feasibility for the collaborative social biking application. Participants reported excellent usability-scores (85 ± 5), high intrinsic motivation in all categories: enjoyment (6.5 ± 0.5), effort/importance (6.4 ± 0.3), relatedness (6.3 ± 0.7) and minimal increase in symptoms of nausea, oculomotor and disorientation. Furthermore, participants found the social aspect enjoyable, agreed that collaboration eased tasks and that they lost track of exercise duration. Interpersonal interaction between participants varied, but was mostly positively rated valence, even if the sense of copresence was limited by physical constraints and avatar representation. Most participants expressed that they would use the program again, but future studies should explore how to improve location and appearance of the virtual coactor, as well as implement additional tasks.

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