1. Placing
Place the six given figures (looking like letters) into the grid so that they do not touch each other, not even diagonally. Numbers below and to the right of the grid show the area occupied by the figures in each corresponding row or column. Figures can be rotated but not reflected. Answer format: write the contents of the diagonal from bottom left to top right. Use letters corresponding to the figure that occupies each cell (even partially) and X for each empty cell. For the example shown, the answer is: DDXOX.
Score: 5 points. | |||||||||||||||||||||||||||

2. Cutting
Cut the given figure onto three parts and reassemble the parts to form a regular hexagon. Parts may be rotated and/or reflected. Answer format: write (in alphabetical order) the nodes which lie along the cut line(s). For the example shown, the answer is: DEGLQ
Score: 8 points. | |||||||||||||||||||||||||||

3. Path
Construct a simple closed loop in the grid which is exactly 44 cells in length. The loop cannot cross or touch itself, not even diagonally. Numbers outside the grid show the number of cells occupied in the corresponding row or column. Answer format: write the contents of the diagonal from bottom left to top right, using 0 for empty cells and 1 for occupied cells. For the example shown, the answer is: 01001.
Score: 6 points. | |||||||||||||||||||||||||||

4. Cube
A cube with a picture on each of its six sides is placed in the top left corner cell of a grid. The cube rolls from cell to cell, visiting each cell exactly once and ending in the bottom right corner cell of the grid. The top view of the cube when it visits some of the cells is shown. Find the cube's path. Answer format:write the number of turns in the cube's path. For the example shown, the answer is: 10.
Score: 8 points. | |||||||||||||||||||||||||||

5. Battleship
The standard fleet (1 four-tube, 2 three-tube, 3 two-tube and 4 one-tube ships) is placed into the grid. Ships cannot touch each other, not even diagonally. A spy satellite took six partial photos of the fleet, but they were shuffled. Locate the positions of all the ships in the grid. All photos except A may be rotated but not reflected. Photo A oriented correctly. Two cells without ships are already marked. Answer format: write the coordinates of the four submarines (one-tube ships). Answer may look like: a1, b2, c3, d4.
Score: 8 points. | |||||||||||||||||||||||||||

6. Words in the honey
Fill the cells with letters so that each of the given words can be read clockwise around one of the black cells. Each word is used exactly once.
Answer format: write the contents of the marked cells from left to right. For the example shown, the answer is: ACAKO.
Score: 7 points. | |||||||||||||||||||||||||||

7. Hexagonal cross-sums
Write the digits from 1 to 9 into the empty cells. Digits in every continuous row (in any direction) are different. Sums for some rows are given Answer format: write the contents of the marked cells (central row) from left to right. Answer may look like: 1234567.
Score: 8 points. | |||||||||||||||||||||||||||

8. Broken calculator
The elements in every digit in the display of a four-digit calculator are scrambled. One element always lights instead of another. In the example shown (multiplication 17*4=68) the digit on the right always has element "b" lighting up instead of "a", "c" instead of "b", "d" instead of "c" and so on. The digit on the left always has "a" lighting instead of "b", "b" instead of "c" and so on. One multiplication is performed on this calculator - the multipliers and product are shown in the picture. Find the correct numbers for this multiplication. Answer format: write the multipliers. For the example shown, the answer is: 17,4.
Score: 7 points. | |||||||||||||||||||||||||||

9. Fly on the polyhedron
A fly sits at the midpoint of one of the edges of a regular polyhedron. The length of the edge is 1 meter. The fly starts his path along a straight line on the polyhedron's surface with the angle between the edge last intersected and the path remaining constant throughout. The path ends when the fly arrives at a vertex or at a face already visited. What is the maximum path length for: À) an octahedron (regular polyhedron with eight faces) B) an icosahedron (regular polyhedron with twenty faces). Sample paths for a tetrahedron and a cube are given. Answer format: write the path's length in centimeters. For the examples shown, the answer could be: A) 216 B) 515.
Score: 5 points for each part. | |||||||||||||||||||||||||||

10. Pipe
By sliding the tiles as in the standard "15 puzzle", construct the longest possible pipe starting at A, using the minimum number of moves. Tile A cannot move. Answer format: first write the pipe's length, then all moves using the letters corresponding to each tile moved. For the example shown, the answer is: 8: HEBCFH.
Score: pipe length minus a quarter of number of moves. | |||||||||||||||||||||||||||

11. 2005 prime
Write a sequence of consecutive prime numbers in increasing order. Using any number of arithmetic signs ( +, - , x, / ) and brackets make an expression with a value equal to 2005. Minimize the difference between maximum and minimum of the prime numbers used. The priority of operation is standard (Multiplication and division before addition and subtraction). 19 - (23 - 29) x 31 = 205The difference between the maximum and minimum numbers equals 12. Answer format: write the expression. For the example shown, the answer is: 19-(23-29)x31.
Score: 20 minus one third of the difference. | |||||||||||||||||||||||||||

12. Rolling maze
A three-dimensional U-pentomino lies in the bottom left corner of a 7x7 grid (as shown in the picture).The pentomino can roll over any one of its edges (but no portion of it can jut out past the grid boundary). The finishing position is marked by the letter F. Place five unit cubes on the grid (aligned with the grid lines) so that the shortest path from the start to the finishing positions has maximum length. For example, with the three unit cubes shown, the shortest path has six moves: two rolls to the "north", two rolls to the "east", and two rolls to the "south". Answer format: first write the number of moves in the shortest path, then the coordinates of the cubes. For the example shown, the answer is: 5: b3, d1, f3.
Score: 0.3 points for every move over 10 plus 1 point if number of moves is correct. |