when a cross tile is placed on a number chart, the number covered by the centre square is the average of all the numbers covered, no matter how large the cross is. Explain why.
2 answers
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We must assume that the centre of the tile is perfectly centered on a square on the number chart, and that the chart is big enough for all of the tile to stay within the borders of the chart.
If this is the case it can be proven in the following way:
Let the number covered by the centre be denoted x. Let y < x be any other number on the chart covered by the cross tile. We may write y = x-k for some natural number k.
Due to symmetry y' = x+k will be covered on the chart as well. Since the tile only covers pairs (x-k,x+k) and each pair has x as average the average of all numbers covered will be x. This could be explained in a little more detail but I hope this suffices as it gets lengthy otherwise!
If this is the case it can be proven in the following way:
Let the number covered by the centre be denoted x. Let y < x be any other number on the chart covered by the cross tile. We may write y = x-k for some natural number k.
Due to symmetry y' = x+k will be covered on the chart as well. Since the tile only covers pairs (x-k,x+k) and each pair has x as average the average of all numbers covered will be x. This could be explained in a little more detail but I hope this suffices as it gets lengthy otherwise!