Form seven-digit numbers that each contain every digit from 1 to 7 exactly once. To do this, start at any one of the seven circles, and proceed from there across the other six circles. In doing so, every circle you enter must be a direct neighbor of the previous circle. The seven-digit number is determined by the sequence of the digits in the circles. How many different seven-digit numbers can be formed in this way?
The ring formed by the six outer circles can be rotated around the central circle into six different positions without altering the visual appearance of the pattern. Consequently, there are only three fundamentally distinct ways to traverse the set of circles.
In the first pattern, you can begin with any of the seven circles and then follow the path in either of two directions. This results in 7 × 6 × 2 = 84 distinct seven-digit numbers. In the second pattern—starting from the top circle—you traverse two outer circles clockwise, then the central circle, and finally four outer circles counterclockwise. The mirror image is also possible: you start at the top circle, traverse two outer circles counterclockwise, go through the central circle and finally traverse four outer circles clockwise. Furthermore, both of these paths can be traversed in reverse. This yields 6 × 2 × 2 = 24 distinct seven-digit numbers. In the final pattern—starting from the top circle—you traverse three outer circles clockwise, then the central circle and finally another three outer circles counterclockwise. Here, too, the mirror image is possible. But because the pattern has 180-degree rotational symmetry, traversing these paths in reverse does not yield any additional numbers. Consequently, this produces only 6 × 2 = 12 distinct seven-digit numbers. In total, using the seven circles, you can generate 84 + 24 + 12 = 120 distinct seven-digit numbers.
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This puzzle originally appeared in Spektrum der Wissenschaft and was reproduced with permission. It was translated from the original German version with the assistance of artificial intelligence and reviewed by our editors.