The box in the photographs uses a Menta without the mint tin (i.e. a demented Menta). However, the code and circuitry will also work with an UNO, Boarduino or any other Arduino compatible microcontroller. If you are using the Menta or Boarduino, first build it according to the appropriate tutorial.
Connect the components as shown in the diagram below.
If you are using a Menta, the pin locations are the same as for the UNO in the diagram, you can mount the MOSFET and Diode and a header for the keypad connector in the prototyping area.
If you are using a Boarduino, be aware that the pin locations will be different from the diagram.