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.
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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!
To understand why the number covered by the center square of a cross-shaped tile on a number chart is the average of all the numbers covered, we need to break down the problem step by step.
Let's suppose we have a number chart, where each cell represents a different number. We place a cross-shaped tile, and the center of the cross overlays a particular number on the chart.
First, let's examine the horizontal arm of the cross. It covers a row of numbers on the chart. To calculate the average of these numbers, we need to add up all the numbers covered by the horizontal arm and then divide the total by the number of covered cells.
Next, let's consider the vertical arm of the cross. Like the horizontal arm, it covers a column of numbers on the chart. We also need to add up all the numbers covered by the vertical arm and divide the total by the number of covered cells to obtain the average of these numbers.
Now, let's observe the center square of the cross, the intersection point of the horizontal and vertical arms. When we calculate the average of the numbers covered by this center square, we essentially add up the numbers from both the horizontal and vertical arms twice (since they overlap at this point). Consequently, to obtain the correct average, we need to divide the sum of the numbers from both arms by the total number of cells.
Since the sums above the horizontal and vertical arms are equal (they represent the same numbers), and each sum is divided by the total number of cells, it follows that the number covered by the center square of the cross will always be the average of all the covered numbers, regardless of the size of the cross-shaped tile.
In summary, the average of the numbers covered by the center square of a cross-shaped tile is obtained by adding up the sums from the horizontal and vertical arms and then dividing by the total number of cells, ensuring all numbers are equally represented in the calculation.