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We consider the problem of digitalizing Euclidean segments. Specifically, we look for a constructive method to connect any two points in Z d . The construction must be consistent (that is, satisfy the natural extension of the Euclidean axioms) while resembling them as much as possible. Previous work has shown asymptotically tight results in two dimensions with Θ(log N ) error, where resemblance between segments is measured with the Hausdorff distance, and N is the L1 distance between the two points. This construction was considered tight because of a Ω(log N ) lower bound that applies to any consistent construction in Z 2 .In this paper we observe that the lower bound does not directly extend to higher dimensions. We give an alternative argument showing that any consistent construction in d dimensions must have Ω(log 1/(d−1) N ) error. We tie the error of a consistent construction in high dimensions to the error of similar weak constructions in two dimensions (constructions for which some points need not satisfy all the axioms). This not only opens the possibility for having constructions with o(log N ) error in high dimensions, but also opens up an interesting line of research in the tradeoff between the number of axiom violations and the error of the construction. In order to show our lower bound, we also consider a colored variation of the concept of discrepancy of a set of points that we find of independent interest.

In this paper we study a cooperative card game called Hanabi from the viewpoint of algorithmic combinatorial game theory. In Hanabi, each card has one among c colors and a number between 1 and n. The aim is to make, for each color, a pile of cards of that color with all increasing numbers from 1 to n. At each time during the game, each player holds h cards in hand. Cards are drawn sequentially from a deck and the players should decide whether to play, discard or store them for future use. One of the features of the game is that the players can see their partners' cards but not their own and information must be shared through hints.We introduce a single-player, perfect-information model and show that the game is intractable even for this simplified version where we forego both the hidden information and the multiplayer aspect of the game, even when the player can only hold two cards in her hand. On the positive side, we show that the decision version of the problem-to decide whether or not numbers from 1 through n can be played for every color-can be solved in (almost) linear time for some restricted cases.