If you understand the rules in “How to Play,” you realize that no particle can be adjacent to the same particle as itself. But each particle must be adjacent to exactly one each of the other two particles. Any other cell that any of those particles is adjacent to must be an empty cell.

I will reference these grid numbers for this puzzle:

|A 1| |B 1| |C 1| |D 1| |E 1|

|A2| |B2| |C2| |D2| |E2|

|A3| |B3| |C3| |D3| |E3|

|A4| |B4| |C4| |D4| |E4|

|A5| |B5| |C5| |D5| |E5|

1.) E5 tells you that one of the following 3 cell combinations must have the three different particles: D4+D5+E4 or D4+D5+C4 or D4+D5+C5 or D3+D4+E4 or D4+E3+E4. So D4 must be a particle and it must be adjacent to the other two. Everything else must be an empty cell. The only cell adjacent to D4 that cannot be a particle is C3, so C3 is X.

2.) C3 says E1 is a particle. This means that either D2 is a particle or both D1 and E2 are particles. Either way, C1, C2, D3, and E3 cannot be particles. All those cells are X.

3.) The only row that can possibly have no particle is the middle row. So A3 and B3 are X.

4.) A3 tells you that A4 or A5 is a neutron. Therefore its electron and proton go in any of these two cells: A4/A5 (whichever the neutron is not in), B4, and B5. This means that C4 and C5 must be X.

5.) A3 also tells you that the neutron that E5 is adjacent to must go in D4. Apply the same logic to the particle in E1. The neutron in that group must be in D2.

6.) You know that A5 and B5 cannot both be electrons. Therefore, according to C5, D5 must be an electron.

|A 1| |B 1| |C 1| |D 1| |E 1|

|A2| |B2| |C2| |D2| |E2|

|A3| |B3| |C3| |D3| |E3|

|A4| |B4| |C4| |D4| |E4|

|A5| |B5| |C5| |D5| |E5|

7.) E4 must be a proton, given the other two adjacent particles.

8.) E4 tells you that B4 cannot be a proton. Since the neutron in that group must be in Column A (according to A3) and the electron must be in Row 5 (according to C5), B4 can be nothing but X.

9.) B4 tells you that there are also four neutrons, since each must be adjacent to exactly one of the other two. Since you know roughly where three electrons are, the only place for another group of particles to go is the top left area, and according to A3, the neutron has to go in B2.

10.) The top right corner is a particle according to C3. The bottom left corner must also be a particle. Therefore, according to B2, another X must go in the top left corner, A1.

11.) The only column that can have three empty cells is Column E, and according to C3, it must go in E2.

12.) According to B2, A5 and E1 must be the same, and both must be particles. Neither can be a neutron since E1 cannot be a neutron. Therefore A4 must be a neutron, according to A3.

|A 1| |B 1| |C 1| |D 1| |E 1|

|A2| |B2| |C2| |D2| |E2|

|A3| |B3| |C3| |D3| |E3|

|A4| |B4| |C4| |D4| |E4|

|A5| |B5| |C5| |D5| |E5|

13.) Columns A and B could easily have one of each particle. Column C does not. Column D could, if D1 is electron. Column E could not. Columns A and B either both have one of each particle, or neither of them does. If neither of them does, then only one column (D) could possibly have one of each. Therefore Columns A and B have one of each particle, and Column D does not. This means D1 cannot be a proton. It cannot be a neutron because it is adjacent to one; therefore D1 must be an electron.

14.) E1 must be a proton.

15.) According to B2, A5 must be a proton, so B5 must be an electron.

16.) According to A4, A2 must be an electron and B1 must be a proton.

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