Puzzle 6

You have 12 coins, one of which is fake. The fake coin is indistinguishable from the rest except that it is either heavier or lighter, but you don't know which. Can you determine which is the fake coin and whether it is lighter or heavier using a balance scale and only 3 weighings?

 [Ref: ZLKY]

Hint: Finding the correct weighings requires some very careful thinking.

Answer: One solution is to label the coins with the letters FAKE MIND CLOT and weigh the coins in the following three combinations:

MA DO -- LIKE
ME TO -- FIND
FAKE  -- COIN

Logic will now allow you to find the fake coin based on the three results. For instance, if the results were left down, balanced, left down, we could work out which coin is fake in the following way:

From the middle weighing we know that the coins METOFIND are all normal. So one of the coins ACKL is fake. Therefore looking at these coins one at a time in the other two weighings, we can see that:

A - appears on the left twice and could be fake.
C - appears only once, therefore can't be fake (otherwise the first weighing would be balanced).
K - appears on opposite sides, so it can't make the left side go down both times.
L - appears only once, therefore can't be fake (otherwise the third weighing would be balanced).

Therefore the only possibility is A, which must be heavier. Any other combination of ups and downs will allow you to use the same logic to find the fake coin.

Puzzle 7

Dodgy Dave did such a good job of the housing estate, he was asked to paint the room numbers on all of the doors of the fourth floor of the local hotel. He painted all of the numbers from 400 to 499. How many times did he paint the number 4?

 [Ref: ZULZ] © Kevin Stone [Protected Puzzle]

Hint: Don't forget rooms 440-449.

Answer: 120 times. Don't forget rooms 440-449, especially room 444.

400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499.

Puzzle 8

Practical Peter was asked to cut a 99 foot rope into three smaller, equal length ropes. However, as usual, Pete couldn't find his measuring tape so he guessed!

When he finally did find his tape (it was under his hat), he discovered that the second piece of rope was twice as long as the first piece minus 35 feet. The third piece of rope was half the length of the first, plus 15 feet. How long were each of the pieces of rope?

 [Ref: ZGCD] © Kevin Stone [Protected Puzzle]

Hint: Only one of the ropes was 33 feet.

Answer:
First  = 34 feet.
Second = 33 feet.
Third  = 32 feet.

This question can be solved easily using algebra, if we call the length of the first rope A, we have:

    Rope 1 = A
    Rope 2 = 2 x A - 35
    Rope 3 = 1 ÷ 2 x A + 15

The three ropes add to 99 feet, so:

    99 = Rope 1 + Rope 2 + Rope 3

    99 = A + (2 x A - 35) + 1 ÷ 2 x A + 15

    99 = 3.5 x A - 20

Adding 20 to both sides we have:

    119 = 3.5 x A

So:

    A = 119 ÷ 3.5

    A = Rope 1 = 34 feet
        Rope 2 = 2 x 34 - 35 = 68 - 35 = 33 feet
        Rope 3 = 1 ÷ 2 x A + 15 = 1 ÷ 2 x 34 + 15 = 17 + 15 = 32 feet

Just to check:

    Rope 1 + Rope 2 + Rope 3 = 34 + 33 + 32 = 99.

As required. QED.

Puzzle 9

How many squares, of any size, can be found on a standard chess board?

How many rectangles, of any size, can be found on a standard chess board? Remember when counting the rectangles, that squares are also rectangles.

 [Ref: ZLYW]

Hint: How many 8x8 squares are there, what about 7x7.

Answer: 1296 rectangles and 204 squares.

Puzzle 10

Last week I travelled from London to Leeds, which is a distance of 174 miles. I started at 9.15am and completed the journey with an average speed of 40 miles per hour. On the way back, in the evening, I travelled exactly the same route, starting at 5.15pm. The traffic was light and I completed the journey with an average speed of 60 miles per hour. What was the overall average speed for round trip? The answer is not 50 mph.