Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?
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.
Three circles have a maximum of six intersections with each other. What is the maximum number of intersections that a hundred circles could have?
Here are the first few sequences from a family of related sequences: $A_0 = 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...$ $A_1 = 2, 6, 10, 14, 18, 22, 26, 30, 34, 38, 42...$ $A_2 = 4, 12, 20, 28, 36, 44, 52, 60...$ $A_3 = 8, 24, 40, 56, 72, 88, 104...$ $A_4 = 16, 48, 80, 112, 144...$ $A_5 = 32, 96, 160...$ $A_6 = 64...$ $A_7 = ...$ . . . Which sequences will contain the number 1000? Once you've had a chance to think about it, click below to see how three different students began working on the task. Alison started by thinking:
Bernard started by thinking:
Charlie started by thinking:
Can you take each of their starting ideas and develop them into a solution? Here are some further questions you might like to consider: How many of the numbers from 1 to 63 appear in the first sequence? The second sequence? ... Do all positive whole numbers appear in a sequence? Do any numbers appear more than once? Which sequence will be the longest? Given any number, how can you work out in which sequence it belongs? How can you describe the $n^{th}$ term in the sequence $A_0$? $A_1$? $A_2$? ... $A_m$?