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Question 1: Can you work out the top number with the bottom left hand corner without working out the middle layers
You can work out the top number if given the number in the bottom left hand corner. You know that the numbers are consecutive on the bottom layer, so you take the sum of the four numbers on the bottom layer of the pyramid and multiply it by two. For example, if the bottom number is 0, you do 0+1+2+3, the numbers in the bottom layer. Then, you multiply this result by 2, and you get the correct answer, 12. This method works with decimals and if the pyramid goes by twos or threes
Question 2: Can you explain why some numbers will never be at the top if you start with a whole number, 0 will NEVER be at the top no matter what whole number is at the bottom. This is because since the pattern is (a+b=c) there is no way for two numbers in the same row to be the same, so you cannot get the number zero at the top. Also, decimals and fractions will never be at the top because you start with integers, so there is no way that a whole number going through the process will be at the top of the pyramid.
Question 3: Can you find a quick way to find the bottom left hand corner number if you are given the top number?
The way I found was to first subtract the top number by four, and then to divide the result by eight and then subtract one. For example, if you have 12 at the top of this pyramid, you can do twelve -four, and then divide by eight, getting 1. you subtract this by one, you get zero, the correct bottom left corner. This pattern does not work for decimals or if it goes by twos or threes. For going up by two, you just divide the top number by eight and subtract one.