Here we have a perfect magic square composed of the numbers I to 16 inclusive. The rows, columns, and two long diagonals all add up 34. Now, supposing you were forbidden to use the two numbers 2 and 15, but allowed, in their place, to repeat any two numbers already used, how would you construct your square so that rows, columns, and diagonals should still add up 34? Your success will depend on which two numbers you select as substitutes for the 2 and 15. 

The four sides of a garden are known to be 20, 16, 12, and 10 rods, and it has the greatest possible area for those sides. What is the area? 

An officer wished to form his men into twelve rows, with eleven men in every row, so that he could place himself at a point that would be equidistant from every row. "But there are only one hundred and twenty of us, sir," said one of the men. Was it possible to carry out the order? 

A man has a square field, 60 feet by 60 feet, with other property, adjoining the highway. For some reason he put up a straight fence in the line of the three trees, as shown, and the length of fence from the middle tree to the tree on the road was just 91 feet. What is the distance in exact feet from the middle tree to the gate on the road? 

A man went into a bank to cash a check. In handing over the money the cashier, by mistake, gave him dollars for cents and cents for dollars. He pocketed the money without examining it, and spent a nickel on his way home. He then found that he possessed exactly twice the amount of the check. He had no money in his pocket before going to the bank. What was the exact amount of that check? 

How many rectangles are there on the chessboard? 

How many squares are there on an 8 x 8 chessboard? 

50 red and 50 blue counters are placed alternately in a line across the floor: RBRBRBR ... RB

By swapping adjacent counters (see arrows) they have to be sorted into 2 groups, with all the reds at one end and all the blues at the other: RRR ... RRRBBB ... BBB

What is the least number of moves needed to do this?

How many moves are needed for n red and n blue counters?

On a calculator you are only allowed to use the keys: 

You can press them as often as you like. You are asked to find a sequence of key presses that produce a given number in the display.

For example, 6 can be produced by "3x4-3-3=".

Find a way of producing each of the numbers from 1 to 10. You must "clear" your calculator before each new sequence.

The number 12 has six factors: 1,2,3,4,6 and 12. Four of these are even (2,4,6 and 12) and two are odd (1 and 3). 

(1) Find some numbers which have all their factors, except 1, even. Describe the sequence of numbers that has this property.

(2) Find some numbers which have exactly half their factors even. Again describe the sequence of numbers that has this property. 

(1) How many cubes are needed to build this tower? 

(2) How would you calculate the number of cubes needed for a tower n cubes high? 

calculate the area of a triangle ABC known that: O(0;0) , A(1;2), B(4;2).

Let x, y be two real numbers such that x + y = 2. Show that x² + y² ≥ 2 and x³ + y³ ≥ 2.

Every day at noon, a ship leaves from Le Havre in France towards New York, and at the same time a ship leaves New York bound to Le Havre. The crossing takes 7 days and 7 nights. If a ship leaves Le Havre today at noon, how many ships coming from New York will it meet? 

 Find the sum of the angles of a five-pointed star?

We are given line segments of lengths 1, 2, 3, . . . , 99. If we have to use all the segments, is it possible to construct a square? How about a rectangle? Justify your answers.

Suppose 4ABC is equilateral, BD/BC=1/3, CE/CA=1/3, and AF/AB=1/3. What is the ratio of the area of the shaded triangle and the area of \(\Delta\)ABC ?

 Let ABCD be a rectangle and let P be a point inside the rectangle. If PA = 8, PB = 4, and PD = 7, then what is PC?

 Twelve points are arranged on a semicircle as shown in the diagram. If every pair of these points is joined by a straight-line segment, then no three of these line segments will intersect at a common point inside the semicircle. How many points are there inside the semicircle where two of these line segments intersect?

The number \(\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}\) equals 

(a) \(\sqrt{5}-1\)      (b) 1     (c) \(\sqrt[3]{2}\)      (d)\(\sqrt{5}-\sqrt[3]{2}\)     (e) \(\dfrac{6}{5}\)  ?