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Use logical deduction to place a different digit from 1 to 9 in each circle below so that 8 of the arrows form the primes 23, 31, 41, 53, 59, 79, 89, and 97. (We view an arrow starting at digit A and ending at digit B as forming the 2-digit number AB.)

A complex arrangement of circles and arrows between them.

Then find a path traversing each arrow exactly once so that the first visit to a circle leaves by its southeast exit, and every subsequent visit leaves by its next exit, going clockwise. What sequence of digits do you get?

Bonus: Extend the sequence by adding the next arrow; inevitably, it will cross over other arrows. Can you rearrange the circles and arrows so that once again no arrows cross? (Pretend we're drawing points and curved lines in the plane—no .)

Hint:

The colors are simultaneously a subtle verification of the first part, a not-so-subtle hint for the bonus, and an Easter egg.

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  • $\begingroup$ You list just 8 two-digit primes, but there are a lot more than 8 arrows. Also, does an arrow go from the left (tens) digit to the right (units) digit? $\endgroup$
    – Jaap Scherphuis
    Mar 14, 2022 at 15:17
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    $\begingroup$ @JaapScherphuis The condition only gives partial information, about 8 of the arrows. And yes, an arrow goes from the left (tens) digit to the right (units) digit. Thanks for asking, I've clarified these points in the question. $\endgroup$
    – noedne
    Mar 14, 2022 at 15:41
  • $\begingroup$ I got the order of the numbers ; I am not able to Proceed because "South East" looks a bit ambiguous. Eg 1 : The centre YELLOW node has East as well as South , hence not sure which comes first. Eg 2 : The centre BLUE has South, East and South East , hence not sure which comes first. I suggest you mark the "Starting Exit" in the Image. $\endgroup$
    – Prem
    Mar 15, 2022 at 19:48
  • $\begingroup$ @Prem Sorry for the confusion, and thanks for the suggestion, but I won't be able to get to it right away. I tried to make it clear by having one arrow start at exactly the southeast of each circle, including the two that you mentioned. $\endgroup$
    – noedne
    Mar 15, 2022 at 21:58

2 Answers 2

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The Numbers can be filled only in this order, top to bottom, left to right:

BLUE 6, 3, 9 ; WHITE 1, 7, 4 ; YELLOW 2, 5, 8 ;

Pictorially:

BLUE 6, 3, 9 ; WHITE 1, 7, 4 ; YELLOW 2, 5, 8 ;

Deductions:

Lots of Possibilities/variations to consider, but basically it goes like this:
The numbers 79 & 97 give a loop; 9 must be the upper right corner BLUE node, because the other node can not form 89;
Now the bottom two YELLOW nodes must contain 5 & 8 to give 59 & 89.
These two Possibilities/variations give two variations to get 31 & 41.
23 eliminates one Possibility/variation.
We are left with the order listed.

Path:

Consider the BLUE node at the upper right corner; It gives a substring (8|5)9793
Consider the YELLOW node at the lower left bottom; It gives a substring 265
Plugging these into google gives PI DIGITS in the results (but user Penguino has already got that)
We can check that the Starting Point it 3 and we get 3.1415926535897932384

Extending:

We can extend with more Pi DIGITS, the next being 6.
When connecting 4 & 6, it will intersect the other connections.
We can avoid that by enlarging the area within 1,3,4,8 and moving those to the outer area and putting all the others in the new inside area.
With this arrangement, we can connect 4 & 6 without intersecting the other connections.

Bonus:

Puzzle was Posted around PI DAY which is great !

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    $\begingroup$ Do you mind adding a drawing of your solution for the bonus when you get the chance? $\endgroup$
    – noedne
    Mar 16, 2022 at 14:22
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    $\begingroup$ @noedne , I do not have the tools right now, but will try that in a little while. $\endgroup$
    – Prem
    Mar 16, 2022 at 15:21
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The numbers are as suggested by Prem. If you start at

3

and follow them in the order shown in the following diagram

enter image description here

you get the first 20 digits of

Pi (3.1415926535897932384)

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