12. 聖保祿學校 On the Moving Sofa Problem The Moving Sofa Problem asks for the maximal area of a twodimensional shape that can be moved around a right-angled corner in a hallway of unit width. We represent the sofa boundary as a parametrized curve to find the integrals that the areamaximised sofa must satisfy. Then, after applying calculus of variations to said integrals to maximise them, we use Green’s theorem to find the equations of the sofa boundary. We also present a computational approach which is able to derive the best shape and area for generalised cases of the problem of non-rightangled hallways. 13. 東華三院盧幹庭紀念中學 Integrality of Generalized Binomial Coefficient We investigate the integrality of one possible generalization of the factorial formula of the well-known binomial coefficients, in which the sum of integers in the denominator is greater than the numerator. We first prove a lower bound for the number of these integral fractions for a given numerator. We then discover one particular pattern of them that is related with the base p-expansion of integers. Finally, we give some corollaries and a graphical representation of them. 14. 華仁書院（九龍） Investigation on the Buffon-Laplace Needle Problem The Buffon-Laplace needle problem is a variation of the wellknown Buffon’s needle problem, which asks what the probability of a needle, after being dropped to a rectangular grid, intersects, or touches the grid is. In this paper, we aim to solve some generalizations of this problem. We generalized the problem by dropping regular polygons to the grid instead of dropping a needle. We solved this generalization of the problem by first using the rotational and reflectional symmetry of the regular polygons, then splitting the number of sides of the polygon into 4 cases, then solving each case. We also generalized the problem by dropping arbitrary 2D shapes. We found a general formula and an algorithmic solution to the problem. Apart from generalizations to the problem, we also considered some variations of the problem, like dropping right regular polygon prisms in a 3D space, with the grid being planes in each axis. We used a similar method to solve this problem and provided a formula. We also considered dropping a needle into a n-dimensional space. However, we failed to get a closed form for the formula.