Daniel Andrade, on his website, has discussed how to program a binary clock with limited equipment in the easiest manner. To make this device, he has only used Arduino, LEDs, resistors, breadboard and buttons. The circuit is also quite simple and he has explained its working and the process of assembling them through videos and diagrams.
The LEDs placed on the breadboard, which is on, are needed to sum together and then it will give the current time. In order to assemble the circuit, firstly the resistors and LEDs are connected together, and then both of them are hooked together from the pin 1 to 13. As the bigger leg of the LED is positive, it is connected to Arduino output pin while the other leg remains on the ground.
Next, to make the code, work on the circuit, the LEDs are used in a particular manner like this, LED 1 to PIN 1, LED 2 to PIN 2, and so on. The buttons are connected using one digital input and two analog inputs and if you want to set change to the hour/minute, two push-buttons are used, connected to the analog input pin 0 and 5.
LEDs are turned ON/OFF by using a normal button, which is connected to digital input pin 0. It works after one leg of the buttons, used on a 2.2K Ohm resistor is connected to the 5V output. The button is put together with the analog/digital input and the other leg goes to the ground.
The code for the clock is inspired from the open-source-arduino-clock by Rob Faludi, with certain modifications. Basically, the time counting remains the same and it provides an opportunity to build another kind of clock, where certain changes in codes are required.
Check out the following link for complete information along with videos and diagrams.