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Problems from Previous Years
 
Problems from 2004

Tray and 9 Triangles 

The tray shown to the right is divided into 9 triangles. Three of the triangles each contain one gold coin. Coins will be added to the tray two at a time and placed, one each, in any two adjacent triangles. You have an unlimited supply of coins. Can you add coins in this manner (two at a time to adjacent triangles) to end with the same number of coins in each triangle?  
Solution 
Tray and 8 Triangles 

The tray shown to the left is divided into 8 triangles. Three of the triangles each contain one gold coin. Coins will be added to the tray two at a time and placed, one each, in any two adjacent triangles. You have an unlimited supply of coins. Can you add coins in this manner (two at a time to adjacent triangles) to end with the same number of coins in each triangle?  
Solution 
Equations I 

The diagram to the right shows four equations,
two across and two down. Two are additions and two are
subtractions. Each number is a twodigit number missing one of
the digits from 1 to 9. Can you fill in the missing digits? 

Solution 
Equations II 

The diagram to the left shows four equations,
two across and two down. Two are additions and two are
subtractions. Each number is a twodigit number missing one of
the digits from 1 to 9. Can you fill in the missing digits? 

Solution 
Rectangles from Squares 

How many rectangles are there in the diagram to the right?
(Remember a square is a rectangle too.) 

Solution 
Rectangles from Squares Squared 

How many rectangles are there in the diagram on the left?  
Solution 
Number Grid 1 

Rearrange the numbers 18 in the squares in the diagram to the right so that the difference between any two horizontally, vertically, or diagonally adjacent numbers is greater than 1. (For example, 2 and 3 are adjacent diagonally and differ by only 1 so the arrangement to the right doesn't work.) 

Solution 
Number Grid 2 

Rearrange the numbers 116 in the squares in the diagram to the left so that the difference between any two horizontally, vertically, or diagonally adjacent numbers is greater than 2. (For example, 4 and 2 are adjacent diagonally and differ by only 2 so the arrangement to the left doesn't work.) 

Solution 
Number Sequence 1 

A teacher writes lists of numbers on a whiteboard starting with a list containing only the number 0. The second list contains 0 and 1, the third contains 0, 1, and 2, and so on. As each number is added to the list it is inserted at a specific position. The diagram to the right shows how the numbers are added. Where should 11 and 12 be inserted in the list, and why? 

Solution 
Number Sequence 2 
What is the next number in the sequence 2, 4, 6, 30, 32, 34, 36, 40, 42, 44, 46, 50, 52, 54, 56, 60, 62, 64, 66, _______ ? 
Solution 
Rook's Tour 1 

A rook's tour is a path through squares of a checkerboard that moves continuously from square to square, moving only horizontally or vertically, and that ends on the square from which it started. The green line in the figure to the right shows a rook's tour. Can you find a rook's tour of this board that does not go through the squares with the symbol and that visits all other squares exactly once? 

Solution 
Rook's Tour 2 

Can you find a rook's tour of the board shown to the left that does not go through the squares with the symbol and that visits all other squares exactly once?  
Solution 
Number Cube 1 

Can you label the edges of a cube with the numbers 1 through 12 so that the sum of the numbers on the edges of all six faces is the same? For example, in the cube to the right, the sum of the edges of the orange face is 29, the blue face is 29, but the purple face sum is 36. 

Solution 
August 2004
September 2004

Four Even's Make an Odd!The four numbers 2, 4, 6, 8 can be combined with parentheses and the symbols for addition, subtraction, multiplication, and division to make a numeric expression in many ways (7680 ways to be exact). See three examples to the left. Despite the fact that the numbers 2, 4, 6, 8 are all even, some of the expressions evaluate to odd numbers. There are 29 expressions using the four numbers 2, 4, 6, 8 combined with the symbols: ( ) +  × ÷ that equal 11 when evaluated. How many can you find? All four numbers must be used exactly once, but may be used in any order. The symbols may be used any number of times (including not at all).  
Solution 

Really Odd  
What is the largest odd number you can make in an expression using the four numbers 2, 4, 6, 8 and the symbols: ( ) +  × ÷ ?  
Solution 
Domino LadderThere are 15 different dominoes that have 0 to 4 pips per square. Can you place those dominoes in the shape of a ladder as shown in the figure to the right so that whenever two dominoes are adjacent, the number of pips in the touching squares is the same? The arrangement shown to the right does not work: the red arrows show adjacent dominoes where the number of pips in the touching squares are not equal. Solution 
Domino RectangleCan you make the rectangle to the left using the 15 dominoes with 0 to 4 pips per square?  
Solution 
How Much is the Red Gem?The picture to the left shows some groups of gems and gold coins that have equal value. For example, two green gems and two red gems is worth the same as four gold coins and one blue gem. How many gold coins is the red gem worth? 

Solution 
How Heavy is the Silver and Gold Plate?A gem merchant has a silver and gold plate that you would like to buy. He will sell it to you for its weight in gold coins. The merchant shows you that one red gem weighs the same as 5 gold coins, a blue gem the same as 7 coins, and a green gem the same as 8 coins. The merchant puts a handful of red gems and one gold coin on a scale and it balances exactly with the silver and gold plate. The merchant then shows you that handful of blue gems and two gold coins balance with the silver and gold plate, and that a handful of green gems with three gold coins also balances with the plate. You are unable to tell how many of each gem are in the handfuls (each handful may have a different number of gems). When you tell the merchant that you have only 100 gold coins, he says, "you have more than enough coins to buy the plate!" What is weight of the silver and gold plate? 

Solution 
 
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