What is the smallest number chart on which this tile can be placed on so its centre squares covers the number 41?
A number chart is a square grid with the smaller squares number consecutively 1,2,3 accross the top row from left to right, then the next row from left to right and so on . A number chart is named after its largest number Eg...
Question : Another tile is a 5 times 5 square of 25 small squares. What is the smallest number chart on which this tile can be placed on so its centre squares covers the number 41?
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The smallest number chart that 41 appears in is a 7-chart. So we restrict our enquiry to 7 and larger.
Note that the center square of a 5-chart is in the 3rd row and 3rd column. Since we are only interested in 7-charts or larger, we do not have to worry about the row placement, so we concentrate on the columns.
In order to cover with the middle square of a 5-chart, 41 must be at least 3 columns away from an edge.
In a 7-chart, the number 41 occurs in the 6th column of the 6th row-- this is next to an edge so it will not work.
In an 8-chart, the number 41 is in the first column or on the edge so this will not work.
In a 9-chart, the number 41 occurs in the 5th column -- this works as there are 4 columns to the left, 4 columns to the right, 4 columns below, and 4 columns above.
I've linked to an article on modular arithmetic -- you will notice that the column placement of the numbers depends on the value of the number in question modulo the size of the chart. `41-= 5(mod9)` so it is in the 5th column. (This means that when you throw out multiples of 9, you are left with a remainder of 5)
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