A chunk is named after the number of squares it contains, or sometimes by its shape. For example, all the dark grey chunks are called 4-chunks and one of them is a 2 by 2 chunk.
The value of a chunk is the sum of the numbers on its squares. For example, the light grey 9-chunk has value (57+58+59+60+69+70+78+79+88=618).
A chunk must be in one piece. It cannot have parts joined at a corner, for example 71,72,83,84. It must not have holes, for example 71,72,73,81,83,91,92,93.
a) On a blank 100-chart, shade in a chunk that has value 600 and the fewest squares. Explain why there is no chunk of value 600 with fewer squares.
b) Explain why every rectangular 15-chunk has a value divisible by 15.
c) Explain why no rectangular 16-chunk has a value divisible by 16.
d) The square 9-chunk [5, 6, 7, 15, 16, 17, 25, 26, 27] has the unusual property that its value 144 is itself a square number. Find the two largest square chunks that have a square value.
sorry idk the answers but i feel your pain with this AMT maths challenge :/
75+85+86+87+88+89+90
Do you know why a 6 chunk cant have a value of 100
Dunno
Dis is so hard
a) To shade in a chunk that has a value of 600 and the fewest squares, we need to find the smallest possible chunk that satisfies this condition. Since the value of a chunk is the sum of the numbers on its squares, we need to look for a combination of numbers on the 100-chart that add up to 600.
To find the chunk with the fewest squares, we need to minimize the number of squares in the chunk. One way to do this is to look for a sequence of consecutive numbers that adds up to 600. Starting from 1, we can keep adding consecutive numbers until we reach a sum close to 600, but not exceeding it.
By trial and error, we can find that the consecutive numbers from 15 to 38 add up to 600. Thus, the chunk [15, 16, 17, ..., 38] has a value of 600 and is the smallest chunk that satisfies this condition.
To explain why there is no chunk of value 600 with fewer squares, we can consider that the smallest possible chunk is a single square. The smallest square in the 100-chart is the number 1. However, the value of this square is only 1. Since we need a chunk with a value of 600, which is much larger than 1, we need to have more squares in the chunk to achieve this value. Therefore, there is no chunk of value 600 with fewer squares than the chunk [15, 16, 17, ..., 38].
b) To explain why every rectangular 15-chunk has a value divisible by 15, we need to consider the properties of rectangular chunks and the definition of divisibility.
A rectangular chunk consists of a grid of squares that form a rectangle shape. In a 100-chart, a rectangular 15-chunk would have dimensions of 3 rows and 5 columns or vice versa (5 rows and 3 columns).
The value of a rectangular chunk is the sum of the numbers on its squares. In this case, the value of a rectangular 15-chunk would be the sum of the 15 numbers within the chunk.
Since all the numbers within the rectangular 15-chunk are consecutive, we can use the formula for the sum of an arithmetic series to find the value. The formula is: sum = (n/2)(first term + last term), where n is the number of terms.
In this case, n = 15 and the first term and last term are consecutive numbers within the chunk. The sum of the rectangular 15-chunk would then be: (15/2)(first term + last term).
Since the number of terms is divisible by 15 (15/15 = 1), and both the first and last terms are consecutive, their sum will always be divisible by 15. Therefore, every rectangular 15-chunk has a value divisible by 15.
c) To explain why no rectangular 16-chunk has a value divisible by 16, we can follow a similar approach as in the previous explanation.
A rectangular 16-chunk consists of a grid of squares with dimensions of 4 rows and 4 columns. The value of the chunk is the sum of the 16 numbers within it.
Using the same formula for the sum of an arithmetic series, we find that the sum of a rectangular 16-chunk would be: (16/2)(first term + last term).
However, in a rectangular 16-chunk, the number of terms (16) is not divisible by 16 (16/16 = 1), unlike in the case of a rectangular 15-chunk. Therefore, the sum of the rectangular 16-chunk will not always be divisible by 16.
Since the sum of the chunk is not divisible by 16, no rectangular 16-chunk will have a value divisible by 16.
d) To find the two largest square chunks that have a square value, we need to consider the given square 9-chunk and look for other square chunks with larger values that are themselves perfect squares.
The given square 9-chunk is [5, 6, 7, 15, 16, 17, 25, 26, 27] with a value of 144, which is equal to 12^2.
To find the largest square chunk, we need to consider chunks that have more squares. One possibility is a square 16-chunk, which would have dimensions of 4 rows and 4 columns. Choosing consecutive numbers from the 100-chart, we can form the square 16-chunk [36, 37, 38, 39, 56, 57, 58, 59, 76, 77, 78, 79, 96, 97, 98, 99]. The value of this chunk is 3600, which is equal to 60^2.
To find the second largest square chunk, we can continue the pattern of increasing the number of squares. A square 25-chunk, with dimensions of 5 rows and 5 columns, would be a suitable choice. Forming the chunk [1, 2, 3, 4, 5, 11, 12, 13, 14, 15, 21, 22, 23, 24, 25, 31, 32, 33, 34, 35, 41, 42, 43, 44, 45], we find that its value is 875, which is equal to 25^2.
Therefore, the two largest square chunks with a square value are the square 16-chunk with a value of 3600 (60^2) and the square 25-chunk with a value of 875 (25^2).