Figure 1: William Friedman
In August of 1940, a team led by William Friedman (See Figure 1), head of the US Army Signal Intelligence Service (SIS), constructed their own purple machine. For the next year and a half, cryptanalysts experimented with the various wires and switches of the wheels of the machine. See the preliminary report on the decryption of the machine written by William Friedman here.
Several months into the experimenting, it was determined that the starting keys for each ten day period were related. Operators changed the key to form that of the next nine days. But this alone was not enough to help the Americans decipher purple.
The Japanese guarded purple carefully. In fact, not a single machine was recovered with all of its components. As a result, not every embassy received a machine. So, in order to send messages to all of the embassies, the Japanese had to send the message out via multiple codes, including those that the Americans had already cracked. This gave the US a crib sheet from which to decode purple messages.
In an effort to gain some useful intelligence, the US staged an electrical failure at the Japanese embassy in Washington DC and two agents from SIS posed as electricians and were able to catch a glimpse of the actual machine. Their accounts helped the SIS team to alter their replica to more closely match the real machine.
The first message was decoded on September 25th, 1940. From that point on, the US was reading fifty to seventy-five messages a day, sometimes even before the Japanese embassy decoded the message.