21 lines
1.2 KiB
Markdown
21 lines
1.2 KiB
Markdown
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Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers
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are all figurate (polygonal) numbers and are generated by the following formulae:
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~~~
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Triangle P3,n=n(n+1)/2 1, 3, 6, 10, 15, ...
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Square P4,n=n2 1, 4, 9, 16, 25, ...
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Pentagonal P5,n=n(3n?1)/2 1, 5, 12, 22, 35, ...
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Hexagonal P6,n=n(2n?1) 1, 6, 15, 28, 45, ...
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Heptagonal P7,n=n(5n?3)/2 1, 7, 18, 34, 55, ...
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Octagonal P8,n=n(3n?2) 1, 8, 21, 40, 65, ...
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~~~
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The ordered set of three 4-digit numbers: `8128, 2882, 8281`, has three interesting properties.
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- The set is cyclic, in that the last two digits of each number is the first two digits of the next number (including the last number with the first).
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- Each polygonal type: triangle (`P(3,127)=8128`), square (`P(4,91)=8281`), and pentagonal (`P(5,44)=2882`), is represented by a different number in the set.
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- This is the only set of 4-digit numbers with this property.
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Find the sum of the only ordered set of six cyclic 4-digit numbers for which each polygonal type:
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triangle, square, pentagonal, hexagonal, heptagonal, and octagonal,
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is represented by a different number in the set.
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