Currently, 91 countries require US citizens to obtain a visa for entry, either as an attachment to their passport or electronically. Some visas (or "tourist cards") may be issued upon arrival and in some cases payment of an accompanying fee.
US State Dept, IATA, Passportinfo.com and Wikipedia consulted. There is conflicting information for certain countries - some airlines require proof of visa - as described by the IATA.
Links to sources:
Passportinfo.com, https://passportinfo.com/countries-that-require-visas-for-us-citizens/ US State Dept, https://travel.state.gov/content/passports/en/country.html and Wikipedia, https://en.wikipedia.org/wiki/Visa_requirements_for_United_States_citizens
The IATA website with detailed passenger/visa information is currently down.
Good catch on Bhutan - I've added it. Iran is complicated for many reasons, but it is possible for a US citizen to secure a visa in advance but when in Iran, US travelers (and Canadians & Brits) must be accompanied by a government minder. Turkey - another complicated story, hopefully tensions will recede soon - hard to believe that NATO allies can wind up with relations this poor. But these days, nothing really shocks me anymore.
Thanks for this great quiz! Could you please add the "n" to Central Africa(n) Republic? I tried it twice and when it didn't work I gave up on that one.