1. Multiple magic square
Fill in the grid so that every row and column contains each number from 1 to 10 exactly once. A triangle will appear between two cells if and only if one number is a multiple of the other, with the point of the triangle indicating the divisor. Answer format: Write the contents of the marked diagonal (form top left to bottom right corner). For the given example the answer would be: 3333.
Score: 5 points. | |

2. Equilibrium of carrousel
The carrousel is a rigid structure which seats form regular decagon. The top view is on the picture. Place 10 weights (from 1 to 10) into the seats according to the following conditions: i. Neighbouring weights have a difference of at least 3. ii. Centre of gravity of all weights is at the centre of the carrousel (the system is in equilibrium). Two weights are already placed. Answer format: Write the numbers starting at the top and going clockwise. Answer may look like: 3,4,5,6,7,8,9,10,1,2.
Score: 5 points. | |

3. Cutting
Cut the given figure into two identical parts. Parts may be rotated and/or reflected. Cut may go only along the sides or the 45-degree diagonals of the squares in the grid. Answer format: First write the number of unit cuts going along the sides of the squares, then write the number of unit cuts going along diagonals. For the given example the answer would be: 3, 2.
Score: 6 points. | |

4. Dividing the strip
Beginning with a strip of length 8 containing the number 36, what is the minimum number of moves required to divide the strip into 8 squares containing the numbers 1 to 8 in order? In one move you may: i) divide a region having more then one cell and number N into two regions each with numbers N/2 if N is even, or numbers (N -1)/2 and (N+1)/2, if N is odd. ii) join two neighbouring regions with numbers M and N producing a new region with the sum of the numbers: M+N. Answer format:Write the minimum number of moves. For the given example the answer would be: 5.
Score: 6 points. | |

5. Mathematical expression
Place the numbers 1, 2, 3, 4 and 5 into the circles and the signs +, -, x, and / between them (every sign and number is used exactly once), so that the resulting mathematical expression is correct from left to right and from right to left. Operations are performed in order they are encountered; multiplication and division have no priority over addition and subtraction. Answer format: Write the numbers from left to right for both expressions. For the given example the answer would be: 52143.
Score: 4 points for each expression. | |

6. How many figures?
Build an octagon with all of its vertices on the nodes of a square grid and with its sides alternating between a single cell side and a cell diagonal. How many different such octagons exist? Figures which differ only in rotation or reflection count as equal. Answer format: Write the number of figures. Answer may look like: 2005.
Score: 7 points. | |

7. Mumbo-jumbo crossword
There are only two letters A and B in the mumbo-jumbo alphabet. The words are arbitrary combinations of A and B. Simple letters is not considered. Fill in the given crossword puzzle with mumbo-jumbo words that are all different. Answer format: Write the four-letter word which was NOT used. Answer may look like: AAAA.
Score: 8 points. | |

8. Grasshopper
A grasshopper starts in S, jumps to the right several times and finishes in F. The lengths of its jumps are given by the number sequence. Whenever the grasshopper arrives at a circle and finds the beginning (tail) of an arrow, it moves to the circle corresponding to the point of the arrow. Draw exactly three arrows to enable the grasshopper to land on F after performing all the jumps specified in the sequence. No arrow can begin or end in S or F. Answer format: Write the cell numbers corresponding to the beginning and end of every arrow. For the given example the answer would be: 6-2, 7-3.
Score: 8 points. | |

9. Pegs
A schoolboy took a few numbered pegs and linked some of them together with bands. His young brother rearranged the pegs and hid their numbers, but the schoolboy was able to determine the numbers of all the pegs by the position of the bands: Now we add an extra twist to this well-known puzzle: The brother not only rearranges the pegs but also removes one of the bands. Construct an arrangement of pegs and bands so that after any rearrangement and removal of any one of the bands, it is possible to determine the numbers of all the pegs. Note that the example above fails to meet these conditions. If we remove band 5-6, then two variants exist: Answer format: First write the number of pegs used, then describe all bands. For the given example the answer would be: 6: 1-2, 1-3, 1-6, 2-3, 3-4, 5-6.
Score: 5 points, plus 3 points for the minimum number of pegs. | |

10. New Year soon!
Using only the following operations, transform 2005 into 2006. i. Write any digit on the right of the number ii. Divide to three (only if number is divisible by 3). The example shows how 25 is transformed into 26 using 19 moves: 25 252 84 28 288 96 32 324 108 36 12 4 45 15 5 54 18 6 2 26 Answer format: First write the number of moves, then all of the additional digits and division signs in the order they were applied. For the given example the answer would be: 19: 2 / / 8 / / 4 / / / / 5 / / 4 / / / 6.
Score: 8 points, plus 2 points for the minimum number of moves. | |

11. Rectangles
Draw a square with side of length N and divide it into DIFFERENT rectangles. For each rectangle calculate the area and the square of the difference of the sides: Answer format: First write your result, R. Then write N, and then describe the sizes of your rectangles (first horizontal size, then vertical) in order by their top left corners, going from top to bottom and from left to right. For the given example the answer would be: 951: 6, 5-2, 1-4, 5-3, 1-2, 5-1.
Score: 15 points for the minimum result, 14 for the next and so on. | |

12. Laser battleship
The sea is a 10x10 grid. A laser beam shines from the middle of the bottom side of the bottom left cell with an angle of 45 degrees, as shown by the arrow. When the beam reaches a ship it is reflected. The path of the beam ends when it reaches the edge of the grid. Place the standard fleet so that the length of the beam will be maximal. Ships may not touch each other, not even diagonally. Answer format: First write the beam's length (assuming that a cell diagonal is of length 1). Then write the coordinates of the top-left cells for all ships going in order of increasing length. Also write the orientation (V-vertical, H-horizontal) for ships with length of more than one cell. For the given example the answer would be: 17.5: a3, e7, g5, c1H, f3V, c5V.
Score: 15 points for the best result, 13 for the next and so on. |