Mathematical & Logical Puzzles -A7

25.     Another Escalator Puzzle

Let the number of visible steps be n and the little boy move up one step in unit time.  Therefore n - 50 steps disappear in 50 units of time.  As the bigger boy travels three times the speed of his little brother, n - 75 steps disappear in 75 ¸ 3 units of time, or 25 units of time.

Therefore:    n - 50  =  n - 75

                       50            25

From this equation, n is readily deduced as 100. Therefore the escalator had 100 steps.

26.     The Ladder Against The Wall

Call AD = a and EC = b.

Since triangle ADF and triangle FEC are similar then:

a  =  4     or  ab = 16...............................................................................................(1)

4      b

AB = a + 4             BC = b + 4.

Therefore by Pythgoras’ Theorem:

AB² + BC² = AC²  or (a + 4)² + (b + 4)² = 400.

Expanding and collecting terms together:

a² + b² + 8(a + b) + 32 = 400..............................................................................(2)

Now from (1)  32 = 2ab

Substituting in (2):

a² + b² + 8(a + b) + 2ab = 400...........................................................................(3)

Now:  a² + 2ab + b² = (a + b)²

So substituting in (3) we get:

(a + b)² + 8(a + b) = 400

Add 16 to both sides:

(a + b)² + 8(a + b) + 16 = 416

Or:  (a + b + 4)² = 416

So: a + b + 4 = Ö416..........................................................................................(4)

From (1)  b = 16

                       a

Substituting in (4):

a + 16 + 4  =  Ö416

       a

By using the formula for solution of quadratic equations, a is found to be 15.36 metres or 1.04 metres.  This makes AB = 19.36 metres or 5.04 metres.

27.     A Coin Puzzle

The rearrangement from Figure 1 to Figure 2 can be effected in four moves.

1.     Move C and D to the right leaving room for another 20c and another 10c between E and C.

2.     Move A and B to the right so that A touches D.

3.     Move D and A into the gap between E and C.

4.     Move E and D into the gap between C and B.

28.     The Dilemma Of A Racecourse Owner

The racecourse owner needs six sets of equipment set out as in the diagram.  A race between any two sets will be a different multiple of 1000 metres.

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