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Choose any three by three square of dates on a calendar page. Circle any number on the top row, put a line through the other numbers that are in the same row and column as your circled number. Repeat this for a number of your choice from the second row. You should now have just one number left on the bottom row, circle it. Find the total for the three numbers circled. Compare this total with the number in the centre of the square. What do you find? Can you explain why this happens?

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Reverse to Order

Take any two digit number, for example 58. What do you have to do to reverse the order of the digits? Can you find a rule for reversing the order of digits for any two digit number?

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Happy Numbers

Take any whole number between 1 and 999, add the squares of the digits to get a new number. Make some conjectures about what happens in general.

Card Trick 2

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

You may have found that this trick does not work if the fourth card from the bottom is the same numerical value and colour as one of the 3 chosen cards or as the first, second or third card from the bottom, a probability of 6/45. When the 'magician' looks at the cards and sees that this has happened the best thing is to carry on with the trick but first to say that the cards should be shuffled again and give some convincing reason!

SUGGESTED SOLUTION

The card which the volunteer keeps will always be the fourth card from the bottom of the pack which has the same numerical value and colour as the predicting card. This is because, whatever 3 cards are selected by the volunteer, with these 3 cards and the predicting card, 4 cards are removed from the pack. Then 45 cards are counted out, and this leaves the last 3 cards to make up 52 altogether. Suppose the 3 cards selected have values x , y and z then the number of cards counted out is (15 - x ) + (15 - y ) + (15 - z ) + x + y + z = 45.

Correct solutions were sent in by:

Sarah - Archbishop Sancroft High School