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Puzzles

17 December

A sequence of zeros and ones can be reduced by writing a 0 or 1 under each pair of numbers: 1 is written if the numbers are the same, 0 is written if they are not. This process can be repeated until there is a single number. For example, if we start with the sequence 1, 1, 1, 0, 1 (of length 5), we get:
1
1
1
0
1
1
1
0
0
1
0
1
0
0
1
The final digit is a 1.
How many sequences of zeros and ones of length 10 are there that when reduced lead to the final digit being a 1?

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21 December

There are 3 ways to order the numbers 1 to 3 so that no number immediately follows the number one less that itself:
Today's number is the number of ways to order the numbers 1 to 6 so that no number immediately follows the number one less that itself.

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13 December

There are 6 ways to split the sequence of the numbers 1 to 5 into three shorter sequences:
Today's number is the number of ways to split the sequence of the numbers 1 to 10 into five shorter sequences.

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18 December

Today's number is the maximum number of pieces that a (circular) pancake can be cut into with 17 straight cuts.

15 December

If the numbers 1 to 7 are arranged 7,1,2,6,3,4,5 then each number is either larger than or a factor of the number before it.
How many ways can the numbers 1 to 7 be arranged to that each number is either larger than or a factor of the number before it?

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