Cisplatin With or Without WEE1 Inhibitor MK-1775 in Treating Patients With Recurrent or Metastatic Head and Neck Cancer

Inclusion Criteria: * Patients must have histologically or cytologically confirmed SCCHN that is recurrent and/or metastatic and not amendable to curative therapy by surgery or radiation; SCCHN originating from the following sites are eligible: oral cavity, oropharynx, larynx, hypopharynx and paranasal sinus; for patients with a diagnosis of SCCHN of unknown origin, their eligibility must be reviewed and approved by the principal investigator * No prior systemic chemotherapy or WEE1 kinase inhibitor therapy for metastatic or recurrent disease will be allowed; patients are permitted to have received prior systemic chemotherapy as a part of the initial multimodality treatment for locally advanced disease if this treatment was completed more than 6 months prior to enrollment * Patients must have disease amenable to biopsy and must be medically fit to undergo a biopsy * Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded) as \> 10 mm with computed tomography (CT) scan; indicator lesions must not have been previously treated with surgery, radiation therapy or radiofrequency ablation unless there is documented progression after therapy * Patients must have completed any previous surgery or radiotherapy \>= 4 weeks prior to enrollment * Eastern Cooperative Oncology Group (ECOG) performance status 0-2 (Karnofsky \> 60%) * Life expectancy of greater than 12 weeks * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1,500/mcL * Platelets \>= 100,000/mcL * Hemoglobin \>= 9 g/dL * Total bilirubin within normal institutional limits * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 × institutional upper limit of normal * Prothrombin time (PT)/partial thromboplastin time (PTT)/international normalized ratio (INR) =\< 1.5 upper limit of normal (ULN) * Creatinine within normal institutional limits OR calculated creatinine clearance \>= 60 mL/min/1.73 m\^2 for patients with levels above institutional normal using modified Cockcroft-Gault * Cardiac function: 12-lead electrocardiogram (ECG) with normal tracing, or non-clinically significant changes that do not require medical intervention; corrected QT (QTc) interval is to be \< 470 msec * Women of childbearing potential and men must be surgically sterilized, practicing abstinence, or agree to use 2 birth control methods prior to study entry and for the duration of study participation including up to 30 days after the last dose of MK-1775; the 2 methods of birth control can be either 2 barrier methods or a barrier method plus a hormonal method to prevent pregnancy; the following are considered adequate barrier methods of contraception: diaphragm or sponge, and condom; other methods of contraception such as copper intrauterine device or spermicide may be used; appropriate hormonal contraceptives will include any registered and marketed contraceptive agent that contains an estrogen and/or a progestational agent (including oral, subcutaneous, intrauterine, or intramuscular agents); should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Past or current malignancy other than SCCHN, except for: * Cervical carcinoma stage 1B or less * Non-invasive basal cell and squamous cell skin carcinoma * Malignant melanoma with a complete response of a duration of \> 10 years * Radically treated prostate cancer (prostatectomy or radiotherapy) with normal prostate specific antigen (PSA), and not requiring ongoing anti-androgen hormonal therapy * Other cancer diagnosis with a complete response of duration of \> 5 years * Patients may not be receiving any other investigational agents * Patients with known brain metastases should be excluded from this clinical trial * History of allergic reactions attributed to compounds of similar chemical or biologic composition to MK-1775 or cisplatin * Patients who are unable to take oral medications and/or who have a clinical or radiological diagnosis of bowel obstruction are ineligible * Uncontrolled intercurrent illness including, but not limited to ongoing or active infection, active peptic ulcer disease, myocardial infarction within 6 months prior to entry, congestive heart failure, symptomatic congestive heart failure, active cardiomyopathy, unstable angina pectoris, cardiac arrhythmia, uncontrolled hypertension or psychiatric illness/social situations that would limit compliance with study requirements * Pregnant women are excluded from this study; breastfeeding should be discontinued if the mother is treated with MK-1775 * Human immunodeficiency virus (HIV)-positive patients are ineligible * Patients taking the following prescription or non-prescription drugs or other products (i.e. grapefruit juice) are ineligible: sensitive cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) substrates, CYP3A4 substrates with a narrow therapeutic index, moderate to potent inhibitors/inducers of CYP3A4; patients would be eligible if the medications can be discontinued two weeks prior to day 1 of dosing and withheld throughout the study until 2 weeks after the last dose of study medication