Question: How many existing basic design types are there of the weapons the United States and the Soviet Union had an arms race over?

Context paragraphs:
Nuclear weapon design: Nuclear weapon designs are physical, chemical, and engineering arrangements that cause the physics package of a nuclear weapon to detonate.  There are three existing basic design types.  In many existing designs, the explosive energy of deployed devices is derived primarily from nuclear fission, not fusion.  (This includes fission of any U-238 tamper or casing due to fusion produced neutrons.)

United States in the Korean War: At the conclusion of World War II the Allied nations began the process of disarmament of Axis controlled regions.  Japan occupied Korea at this time and had been in control since 1910.  In 1945, the decision was made to have American Marines forces oversee Japanese surrender and disarmament south of the 38th parallel and the Soviet Union would facilitate the change of power to the north.  At the time there was no political motivation and seemed to be a logical and convenient plan of action.  The original agreement and intent was to create a unified and independent Korea out of the post Japanese occupation era.  Instead each side of the 38th parallel established its own government under the influence of the occupational country; the United States in South Korea and the Soviet Union in North Korea.  Both new Korean governments discredited the other and claimed to be the only legitimate political system.  Tensions between the North and South escalated and each side began to petition foreign powers for resources and support.  South Korea wanted weapons and supplies from Truman and the United States government while North Korea sought help from Stalin and the Soviet Union.  The United States was still war weary from the disruptive World War II campaign and refused South Korea's request for weapons and troops.  North Korea convinced the Soviet Union to supply them with the weapons and support they requested.  This decision coincided with the United States withdrawing the last remaining combat troops from South Korea.  North Korea saw its opportunity and attacked South Korean forces at the 38th parallel on June 25, 1950 and thus initiating the Korean War.

List of nuclear weapons tests of the Soviet Union: The nuclear weapons tests of the Soviet Union were performed between 1949 and 1990 as part of the nuclear arms race.  The Soviet Union conducted 715 nuclear tests using 969 total devices by official count, including 219 atmospheric, underwater, and space tests and 124 peaceful use tests.  Most of the tests took place at the Southern Test Site in Semipalatinsk, Kazakhstan and the Northern Test Site at Novaya Zemlya.  Other tests took place at various locations within the Soviet Union, including now-independent Kazakhstan, Uzbekistan, Ukraine and Turkmenistan.

American University speech: The American University speech, titled "A Strategy of Peace", was a commencement address delivered by President John F. Kennedy at the American University in Washington, D.C., on Monday, June 10, 1963.  Delivered at the height of his rhetorical powers and widely considered one of his most powerful speeches, Kennedy not only outlined a plan to curb nuclear arms, but also "laid out a hopeful, yet realistic route for world peace at a time when the U.S. and Soviet Union faced the potential for an escalating nuclear arms race."  In the speech, Kennedy announced his agreement to negotiations "toward early agreement on a comprehensive test ban treaty" (which resulted in the Nuclear Test-Ban Treaty) and also announced, for the purpose of showing "good faith and solemn convictions", his decision to unilaterally suspend all US atmospheric testing of nuclear weapons as long as all other nations would do the same.  Noteworthy are his comments that the United States was seeking a goal of "complete disarmament" of nuclear weapons and his vow that America "will never start a war".  The speech was unusual in its peaceful outreach to the Soviet Union at the height of the Cold War, and is remembered as one of Kennedy’s finest and most important speeches.

Soviet influence on the peace movement: During the Cold War (1947–1991), when the Soviet Union and the United States were engaged in an arms race, the Soviet Union promoted its foreign policy through the World Peace Council and other front organizations.  Some writers have claimed that it also influenced non-aligned peace groups in the West, although the CIA and MI5 have doubted the extent of Soviet influence.

Nunn–Lugar Cooperative Threat Reduction: As the fall of the Soviet Union appeared eminent, the United States and their allies began to worry about the concept that the nuclear weapons held in smaller countries by the Soviet Union could fall or would fall into enemy hands.  The Cooperative Threat Reduction (CTR) Program was an initiative housed within the Defense Threat Reduction Agency (DTRA).  The CTR program is better known as the Nunn–Lugar Act based on the Soviet Nuclear Threat Reduction Act of 1991 which was authored and cosponsored by Sens.  Sam Nunn (D-GA) and Richard Lugar (R-IN).  This Act was created in 1986 in a congressional meeting.  According to the CTR website, "the purpose of the CTR Program is to secure and dismantle weapons of mass destruction and their associated infrastructure in former Soviet Union states. "Another explanation of purpose of the program is "to secure and dismantle weapons of mass destruction in states of the former Soviet Union and beyond".

Nuclear arms race: The nuclear arms race was a competition for supremacy in nuclear warfare between the United States, the Soviet Union, and their respective allies during the Cold War.  During this period, in addition to the American and Soviet nuclear stockpiles, other countries developed nuclear weapons, though none engaged in warhead production on nearly the same scale as the two superpowers.

Cold War (1985–91): The Cold War period of 1985–1991 began with the rise of Mikhail Gorbachev as leader of the Soviet Union.  Gorbachev was a revolutionary leader for the USSR, as he was the first to promote liberalization of the political landscape (Glasnost) and capitalist elements into the economy (Perestroika); prior to this, the USSR had been strictly prohibiting liberal reform and maintained an inefficient centralized economy.  The USSR, despite facing massive economic difficulties, was involved in a costly arms race with the United States under President Ronald Reagan.  Regardless, the USSR began to crumble as liberal reforms proved difficult to handle and capitalist changes to the centralized economy were badly transitioned and caused major problems.  After a series of revolutions in Soviet Bloc states, and a failed coup by conservative elements opposed to the ongoing reforms, in New Years Eve the Soviet Union collapsed in 1991 and the Cold War came to an end.

Aftermath of World War II: The aftermath of World War II was the beginning of an era defined by the decline of the old great powers and the rise of two superpowers: the Soviet Union (USSR) and the United States of America (USA).  Allied during World War II, the USA and the USSR became competitors on the world stage and engaged in the Cold War, so called because it never resulted in overt, declared hot war between the two powers but was instead characterized by espionage, political subversion and proxy wars. Western Europe and Japan were rebuilt through the American Marshall Plan whereas Eastern Europe fell under the Soviet sphere of influence and eventually an "Iron Curtain".  Europe was divided into a US-led Western Bloc and a Soviet-led Eastern Bloc.  Internationally, alliances with the two blocs gradually shifted, with some nations trying to stay out of the Cold War through the Non-Aligned Movement.  The Cold War also saw a nuclear arms race between the two superpowers; part of the reason that the Cold War never became a "hot" war was that the Soviet Union and the United States had nuclear deterrents against each other, leading to a mutually assured destruction standoff.

The Bomb (film): The Bomb is a 2015 American documentary film about the history of nuclear weapons, from theoretical scientific considerations at the very beginning, to their first use on August 6, 1945, to their global political implications in the present-day.  The two-hour PBS film was written and directed by Rushmore DeNooyer, who noted the project took a year and a half to complete, since much of the film footage and images was only recently declassified by the United States Department of Defense.  According to DeNooyer, “It wouldn’t take very many bombs to really change life on Earth, ... The idea that there are thousands of them sitting around is pretty scary.  I don’t think people today realize that.  They don’t think about it.  I don’t think they are scared.  But in a way, they should be.”  Mark Dawidziak, of the "Cleveland Plain Dealer", summarized the film as follows: ""The Bomb" moves swiftly to cover Hiroshima and Nagasaki, the Cold War, the arms race, the Red Scare, the witch hunt, the Cuban Missile Crisis, test-ban treaties, the "Star Wars" initiative, the anti-nuke movement, the collapse of the Soviet Union and the rise of new nuclear threats."  According to historian Richard Rhodes, “The invention [of 'The Bomb'] was a millennial change in human history: for the first time, we were now capable of our own destruction, as a species.”

Supporting paragraphs: Nuclear arms race Nuclear weapon design
Answer: three