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Number Triangle
Hi I am wondering if someone could help me on this
Here are two examples of a number triangle. There are four consecutive positive integers 10, 11, 12 and 13, on the bottom line, in some order. Starting from the left, each number in the bottom line is added to the number next to it and the answer written above the space between them. This is repeated for each line, until there is a single number on top.
Please see attached figure:
Number triangle with different numbers in the bottom line, or with the same numbers but in a different order, are regarded as different. The two examples shown are, there different.
a. By changing the arrangements of the numbers 10, 11, 12 and 13 in the bottom line, what top numbers are possible?
b. Another number triangle has six consecutive integers in some order in the bottom line and top number 2006.
i. Give two examples of this: one with 60, 61, 62, 63, 64 and 65, in some order, in the bottom line, and the other with 61, 62, 63, 64, 65 and 66, in some order, in the bottom line.
ii. Show that these are the only possible sets of six consecutive numbers for the bottom line of a number triangle with top number 2006.
iii. How many different number triangles are there with six consecutive numbers in the bottom line, in some order, and with top number 2006?(Recall that two number triangles are different if their bottom line numbers are different or their different bottom line orders are different).