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Known are five numbers: 1, 2, 3, 4, and 5, arranged clockwise. The number after 1 is 2, after 2 is 3, after 3 is 4, after 4 is 5, and after 5 is 1. Whenever two adjacent numbers meet, it triggers a foldback to the digit before the starting number. For example, when 1 and 2 meet, it triggers a foldback, going from 2 back to 1, and then from 1 back to 5, denoted as 1☺2=5; when 2 and 3 meet, it triggers a foldback, going from 3 back to 2 and then from 2 back to 1, denoted as 2☺3=1; and so on. We have the following equations:

1 ☺ 2 = 5

2 ☺ 3 = 1

3 ☺ 4 = 2

4 ☺ 5 = 3

5 ☺ 1 = 4

Based on this, we have the following rules:

(1) 1☺1=4, 2☺2=5, 3☺3=1, 4☺4=2, 5☺5=3

(2) When the digit 3 appears after the operator ☺, ▲3 becomes ♠3 or ♥3, and ▽3 becomes ♦3 or ♣3. ♠3=5, ♣3=4, ♥3=2, ♦3=1. Only the digit 3 can be represented by ♠, ♣, ♥, or ♦; other digits can only be represented by ▲ or ▽.

(3) ▲ can only meet ▲, ♠, or ♥; ▽ can only meet ▽, ♦, or ♣.

(4) ▽1, ▽2, ▽3, ▽4, ▽5 cannot have the same value; ▲1, ▲2, ▲3, ▲4, ▲5 cannot have the same value.

The following equations are all true:

▽1☺▽2=2

▽2☺♦3=▽2☺♣3=5

▽3☺▽4=3

▽4☺▽5=4

▽5☺▽1=4 ▲1☺▲2=4

▲2☺♠3=▲2☺♥3=3

▲3☺▲4=3

▲4☺▲5=3

▲5☺▲1=4

▽3☺▽2=1

▽4☺▽2=4

▽2☺▽1=2

▲4☺▲2=3

Now, let's find the values of the unknowns:

▽1 = ? ▽2 = ? ▽3 = ? ▽4 = ? ▽5 = ? ▲1 = ? ▲2 = ? ▲3 = ? ▲4 = ? ▲5 = ?

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  • $\begingroup$ It doesn't seem to have a correct solution. $\endgroup$
    – Nautilus
    Commented Jul 31, 2023 at 8:43
  • $\begingroup$ Does ▽5☺▽1=4 ▲1☺▲2=4 mean ▽5☺▽1=4 and ▲1☺▲2=4? $\endgroup$
    – Nautilus
    Commented Jul 31, 2023 at 8:58

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