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→Решение XOR-SAT задачи методом Гаусса
| ("⊕" means XOR, the <font color=red> red clause </font> is optional)
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| (''a''⊕''c''⊕''d'') ∧ (''b''⊕¬''c''⊕''d'') ∧ (''a''⊕''b''⊕¬''d'') ∧ (''a''⊕¬''b''⊕¬''c'') {{<font color|#ff8080|=red> ∧ (¬''a''⊕''b''⊕''c'')}}</font>
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| colspan="9" | Each clause leads to one equation.
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| || ''a'' || ⊕ || || ''c'' || ⊕ || || ''d'' || = 1
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| || ''b'' || ⊕ || ¬ || ''c'' || ⊕ || || ''d'' || = 1
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| || ''a'' || ⊕ || || ''b'' || ⊕ || ¬ || ''d'' || = 1
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| || ''a'' || ⊕ || ¬ || ''b'' || ⊕ || ¬ || ''c'' || = 1
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| {{<font color|#ff8080|=red> ¬}} || {{</font> <font color|#ff8080|=red> ''a''}} || {{</font> <font color|#ff8080|=red> ⊕}} || || {{</font> <font color|#ff8080|=red> ''b''}} || {{</font> <font color|#ff8080|=red> ⊕}} || || {{</font> <font color|#ff8080|=red> ''c''}} || {{</font><font color|#ff8080| =red> ≃ 1}}|}</font>
|-
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| ''a'' || ⊕ || ''b'' || ⊕ || ''c'' || = '''1'''
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| {{<font color|#ff8080|=red> ''a''}} </font> || {{<font color=red> ⊕ </font> |#ff8080|⊕}} || {{<font color|#ff8080|=red> ''b''}} </font> || {{<font color=red> ⊕ </font> |#ff8080|⊕}} || {{<font color|#ff8080|=red> ''c''}} </font> || {{<font color|#ff8080| =red> ≃ '''0'''}}</font>
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| colspan="6" | (If the {{<font color|#ff8080|=red> red equation}} </font> is present, {{<font color|#ff8080|=red> it}} </font> contradicts
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| colspan="6" | the last black one, so the system is unsolvable.
! Solution:
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| If the {{<font color|#ff8080|=red> red clause}} </font> is present: || Unsolvable
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| Else: || ''a'' = 0 = FALSE
| colspan="2" | '''As a consequence:'''
|-
| colspan="2" | ''R''(''a'',''c'',''d'') ∧ ''R''(''b'',¬''c'',''d'') ∧ ''R''(''a'',''b'',¬''d'') ∧ ''R''(''a'',¬''b'',¬''c'') {{<font color|#ff8080|=red> ∧ ''R''(¬''a'',''b'',''c'')}}</font>
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| colspan="2" | is not 1-in-3-satisfiable,
