There are n bags with coins that look the same. Each bag has an infinite number of coins and all coins in the same bag weigh the same amount. Coins in different bags weigh 1, 2, 3, and so on to n grams exactly. There is a unique label from the set 1 through n attached to each bag that is supposed to correspond to the weight of the coins in that bag. The task is to confirm all the labels by using a balance scale once.We study weighings that we call downhill: they use the numbers of coins from the bags that are in a decreasing order. We show the importance of such weighings. We find the smallest possible total weight of coins in a downhill weighing that confirms the labels on the bags. We also find bounds on the smallest number of coins needed for such a weighing.Example. On the right pan, place one coin from the bag labeled 2, two coins from the bag labeled 3, and so on such that there are i − 1 coins from the bag labeled i. On the left pan, put coins from the bag labeled 1 to match the right pan in total weight. For example, if n = 4, we put 20 coins labeled 1 on the left pan and coins with the labels 2, 3, 3, 4, 4, and 4 on the right pan. Any rearrangement of the bags makes the left pan heavier or the right pan lighter. Thus, all the labels are confirmed if the scale balances.Our first goal in this paper is to find a weighing that minimizes the total weight on both pans. Our second goal is to minimize the number of coins used.In Section 2, we provide definitions, examples, and basic results. We study downhill weighings, which are weighings that have decreasing multiplicities, that is weighings that use fewer coins of weight i than weight j, where i > j. For consistency, we assume that the multiplicities of the coins on the right pan are negative. In Section 3, we explain the idea of a separation point and how it helps to calculate the minimum weight for a downhill weighing. In Sections 4, 5, and 6, we calculate the minimum weight explicitly depending on the remainder of n modulo 3. Section 4 is devoted to the simplest case of n = 3k + 1. In this case, the minimum weight is 8n 3 +12n 2 −12n−8 81. In Section 5 we study the case of 3k and find the minimum weight to be 8n 3 +27n 2 +9n−81

A magic SET square is a 3 by 3 table of SET cards such that each row, column, diagonal, and anti-diagonal is a set. We allow the following transformations of the square: shuffling features, shuffling values within the features, rotations and reflections of the square. Under these transformations, there are 21 types of magic SET squares. We calculate the number of squares of each type. In addition, we discuss a game of SET tic-tac-toe.

A magic SET square is a 3 by 3 table of SET cards such that each row, column, diagonal, and anti-diagonal is a set. We allow the following transformations of the square: shuffling features, shuffling values within the features, rotations and reflections of the square. Under these transformations, there are 21 types of magic SET squares. We calculate the number of squares of each type. In addition, we discuss a game of SET tic-tac-toe. The game of SETThe game of SET is one of the most mathematical games. To play it you need a deck of special cards. Each card has one, two, or three identical objects drawn on it. The objects could be of three different colors: red, green, or purple. The object can have three different shapes: oval, diamond, and squiggly. The cards can also differ in shading. There are three types of shading: empty, full, and striped.Overall, there are four features: number, shape, color, and shading. And there are three possibilities for each feature. The total number of cards in the deck is 81. Each card has a unique choice for one of the four features.Three cards can form a set if and only if for every feature all three cards are either the same or all different. Figure 1 shows an example of a set.All three cards have the same number of objects-1, all the cards have the same shadingstriped. All the cards have different colors and different shapes.