Schools Mathematics Grand Challenge
Two things to remember for this week's problems:
 If a circle has radius R, then its circumference is 2 x pi x R;
 If a circle has radius R, then its area is pi x R x R.
PROBLEM 13:
A conference centre contains five identical rooms. All of the rooms have
square floors. Recently, the floors of the rooms have been painted, and a
different pattern has been used for each room. Each room has a certain
number of red circles painted on a white background in the patterns
that you can see by clicking here.
 Room 1 has 9 red circles of equal area painted on a white background
 Room 2 has 16 red circles of equal area painted on a white background;
 Room 3 has 25 red circles of equal area painted on a white background;
 Room 4 has 36 red circles of equal area painted on a white background;
 Room 5 has 49 red circles of equal area painted on a white background;
Which room has the largest redpainted area on its floor? If one room
has a greater redpainted area than the rest, give the number of that
room (between 1 and 5 inclusive) as your answer. If all of the rooms
have the same redpainted area, give 0 as your answer.
(You can ignore the black borders in the picture when figuring out the areas.)
PROBLEM 14:
For this problem, you should use the value 3.14 as an approximation for pi.
A communications company wanted to put a telephone cable around the
equator of the earth. They had planned to place the cable along the
surface but, due to a miscalculation, they ordered a cable that was 34.54m
too short to reach all the way around. After much headscratching,
they decided that the best thing to do was to dig a trench of a fixed
depth (same depth all the way around the equator), and place the cable
at the bottom of the trench. After doing this, the cable was exactly
the right length to fit once around the earth. Assuming that the equator
is a circle, how deep in metres was the trench? Give your answer correct
to one decimal place.
PROBLEM 15:
Mary has a small pile of 14 cards, which contains 13 black cards and one
red card. She deals the cards in the following way. First, she takes
the card on top of the pack and places it on a table. The next card
(the one on top of the remaining 13 cards) is then moved to the bottom of
the pack. She repeats this process, placing a card on the table and then
moving the next card to the bottom of the pack, until all of the 14 cards
have been placed on the table. If the 12th card placed on the table is
the red card, what was the position of the red card in the original pack?
Give your answer as a number between 1 and 14 (inclusive), where 1 means
the top card, 2 the card second from the top, 3 the card third from the
top and so on.


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