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  #### Resources tagged with Generalising similar to All Square:
  
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  ### Nim-7
  
  ##### Stage: 1, 2 and 3 Challenge Level:
  
  Can you work out how to win this game of Nim? Does it matter if you go first or second?
  
  ### Maths Trails
  
  ##### Stage: 2 and 3
  
  The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.
  
  ### More Magic Potting Sheds
  
  ##### Stage: 3 Challenge Level:
  
  The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?
  
  ### Cubes Within Cubes Revisited
  
  ##### Stage: 3 Challenge Level:
  
  Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?
  
  ### Consecutive Negative Numbers
  
  ##### Stage: 3 Challenge Level:
  
  Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?
  
  ##### Stage: 3 Challenge Level:
  
  Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is... When was he born?
  
  ### Cunning Card Trick
  
  ##### Stage: 3 Challenge Level:
  
  Delight your friends with this cunning trick! Can you explain how it works?
  
  ### Tourism
  
  ##### Stage: 3 Challenge Level:
  
  If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.
  
  ### Mini-max
  
  ##### Stage: 3 Challenge Level:
  
  Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .
  
  ### Where Can We Visit?
  
  ##### Stage: 3 Challenge Level:
  
  Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?
  
  ### What's Possible?
  
  ##### Stage: 4 Challenge Level:
  
  Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?
  
  ### Picturing Triangular Numbers
  
  ##### Stage: 3 Challenge Level:
  
  Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
  
  ### Card Trick 2
  
  ##### Stage: 3 Challenge Level:
  
  Can you explain how this card trick works?
  
  ### Summing Consecutive Numbers
  
  ##### Stage: 3 Challenge Level:
  
  Many numbers can be expressed as the sum of two or more consecutive integers. For example, 15=7+8 and 10=1+2+3+4. Can you say which numbers can be expressed in this way?
  
  ### Chocolate Maths
  
  ##### Stage: 3 Challenge Level:
  
  Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .
  
  ### Have You Got It?
  
  ##### Stage: 3 Challenge Level:
  
  Can you explain the strategy for winning this game with any target?
  
  ### Handshakes
  
  ##### Stage: 3 Challenge Level:
  
  Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?
  
  ### Chess
  
  ##### Stage: 3 Challenge Level:
  
  What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?
  
  ### Christmas Chocolates
  
  ##### Stage: 3 Challenge Level:
  
  How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?
  
  ### Squares in Rectangles
  
  ##### Stage: 3 Challenge Level:
  
  A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?
  
  ### Sum Equals Product
  
  ##### Stage: 3 Challenge Level:
  
  The sum of the numbers 4 and 1 [1/3] is the same as the product of 4 and 1 [1/3]; that is to say 4 + 1 [1/3] = 4 × 1 [1/3]. What other numbers have the sum equal to the product and can this be so for. . . .
  
  ### Special Sums and Products
  
  ##### Stage: 3 Challenge Level:
  
  Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.
  
  ### Picturing Square Numbers
  
  ##### Stage: 3 Challenge Level:
  
  Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?
  
  ### Shear Magic
  
  ##### Stage: 3 Challenge Level:
  
  What are the areas of these triangles? What do you notice? Can you generalise to other "families" of triangles?
  
  ### Pair Products
  
  ##### Stage: 4 Challenge Level:
  
  Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?
  
  ### Steps to the Podium
  
  ##### Stage: 2 and 3 Challenge Level:
  
  It starts quite simple but great opportunities for number discoveries and patterns!
  
  ### Magic Letters
  
  ##### Stage: 3 Challenge Level:
  
  Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?
  
  ### Route to Infinity
  
  ##### Stage: 3 Challenge Level:
  
  Can you describe this route to infinity? Where will the arrows take you next?
  
  ##### Stage: 3 Challenge Level:
  
  List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?
  
  ### Pick's Theorem
  
  ##### Stage: 3 Challenge Level:
  
  Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.
  
  ### Threesomes
  
  ##### Stage: 3 Challenge Level:
  
  Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?
  
  ### Enclosing Squares
  
  ##### Stage: 3 Challenge Level:
  
  Can you find sets of sloping lines that enclose a square?
  
  ### Converging Means
  
  ##### Stage: 3 Challenge Level:
  
  Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .
  
  ### What Numbers Can We Make Now?
  
  ##### Stage: 3 Challenge Level:
  
  Imagine we have four bags containing numbers from a sequence. What numbers can we make now?
  
  ### Lower Bound
  
  ##### Stage: 3 Challenge Level:
  
  What would you get if you continued this sequence of fraction sums? 1/2 + 2/1 = 2/3 + 3/2 = 3/4 + 4/3 =
  
  ### Pareq Calc
  
  ##### Stage: 4 Challenge Level:
  
  Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .
  
  ### Steel Cables
  
  ##### Stage: 4 Challenge Level:
  
  Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?
  
  ### Go Forth and Generalise
  
  ##### Stage: 3
  
  Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.
  
  ### More Number Pyramids
  
  ##### Stage: 3 Challenge Level:
  
  When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...
  
  ### How Much Can We Spend?
  
  ##### Stage: 3 Challenge Level:
  
  A country has decided to have just two different coins, 3z and 5z coins. Which totals can be made? Is there a largest total that cannot be made? How do you know?
  
  ### More Twisting and Turning
  
  ##### Stage: 3 Challenge Level:
  
  It would be nice to have a strategy for disentangling any tangled ropes...
  
  ### Egyptian Fractions
  
  ##### Stage: 3 Challenge Level:
  
  The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.
  
  ### Partitioning Revisited
  
  ##### Stage: 3 Challenge Level:
  
  We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4
  
  ### Cuboid Challenge
  
  ##### Stage: 3 Challenge Level:
  
  What size square corners should be cut from a square piece of paper to make a box with the largest possible volume?
  
  ### For Richer for Poorer
  
  ##### Stage: 3 Challenge Level:
  
  Charlie has moved between countries and the average income of both has increased. How can this be so?
  
  ### Keep it Simple
  
  ##### Stage: 3 Challenge Level:
  
  Can all unit fractions be written as the sum of two unit fractions?
  
  ### All Tangled Up
  
  ##### Stage: 3 Challenge Level:
  
  Can you tangle yourself up and reach any fraction?
  
  ### Squares, Squares and More Squares
  
  ##### Stage: 3 Challenge Level:
  
  Can you dissect a square into: 4, 7, 10, 13... other squares? 6, 9, 12, 15... other squares? 8, 11, 14... other squares?
  
  ### Number Pyramids
  
  ##### Stage: 3 Challenge Level:
  
  Try entering different sets of numbers in the number pyramids. How does the total at the top change?
  
  ### Searching for Mean(ing)
  
  ##### Stage: 3 Challenge Level:
  
  Imagine you have a large supply of 3kg and 8kg weights. How many of each weight would you need for the average (mean) of the weights to be 6kg? What other averages could you have?
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