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### Schools Mathematics Grand Challenge

Two things to remember for this week's problems:

1. If a circle has radius R, then its circumference is 2 x pi x R;
2. 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 red-painted area on its floor? If one room has a greater red-painted 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 red-painted 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 head-scratching, 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.