Two of the (pairs of) faces of the cuboids whose sides are in the 4:2:1 ratio are dominoes, so as a little homage to Lewis Carroll, these double dominoes could be called “dodominoes” (Charles Dodgson used to stutter when say his last name: Do… Do… Dodgson, so he represented himself, in Alice in Wonderland, like a dodo bird). Therefore, our ideal tetrabrik for the last few weeks is a versatile 20x10x5 cm dodomino.
This versatility allows tetrabriks to cover a standard 40 cm side chess board in many different ways, if we can support them on any of their faces. To begin with, we can choose from 81 different combinations of faces, from 8 faces of 20×10 to 32 faces of 10×5. These 81 combinations are the solutions of the Diophantine equation 8x + 4y + 2z = 64 (or what is the same, 4x + 2y + z = 32), relative, for simplicity, to 4x2x1 dodominoes on an 8×8 board, where x, y, z are, respectively, the numbers of each type of faces in each of the possible combinations.
In turn, each combination of faces allows a certain number of different coatings. For example, with 8 sides of 4×2 we can cover the board in 9 different ways, as shown in the image sent by Salva Fuster.
The constructions with 3D dodominoes raised in previous weeks still do not arouse the interest of my astute readers (because I do not think they exceed their analytical capacities), so the question remains open.
Alquerque
No. 3122
S2312
E 3132
or 2213
The 4×4 board in the figure represents a block of buildings, one per square. In each line, horizontal or vertical, the buildings are all of different height, and the heights vary between 10 and 40 meters. The attached table indicates how many buildings are visible from each direction. For example, if we looked at the sequence of heights 10, 40, 30, 20 from left to right we would see 2 buildings (those with heights 10 and 40 meters) and looking from right to left we would see 3 (those with heights 20, 30 and 40 meters). meters). The table lists the number of buildings viewed from each side (North, South, East, and West) in each column and row (columns are ordered from right to left and rows from top to bottom). So, 0 2213 means that from the left side of the grid we see 2 buildings in the first row, 2 in the second, 1 in the third, and 3 in the fourth. What is the layout of the buildings?
This problem has little to do, in principle, with our dodominoes; but the Alquerque Group of Seville – a commendable team of mathematics teachers concerned with making problem solving more attractive and understandable for students – used 4x2x1 cm wooden blocks to physically solve this and similar problems.
By the way, do you know what the alquerque is and what relationship it has with the boards of nxn squares?
Carlo Frabetti is a writer and mathematician, member of the New York Academy of Sciences. He has published more than 50 popular science works for adults, children and young people, including ‘Damn Physics’, ‘Damn Mathematics’ or ‘The Great Game’. He was a screenwriter for ‘The Crystal Ball’.
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