Yeah you'd need 8 battery holders, 1 was just included in the "dev kit" option I linked already (as was 1 picaxe)
And yeah the button you linked is fine
You can check their distributors here: Distributors - PICAXE
US options from SparkFun:
Proto board (which you'll use to program them): https://www.sparkfun.com/products/8321
USB to Serial cable: https://www.sparkfun.com/products/8312
Battery holder: https://www.sparkfun.com/products/10891
10K resistor pack: https://www.sparkfun.com/products/11508
(I think this'll be ok, double check values)
MOSFET symbol: https://www.google.com.au/search?q=m...w=1920&bih=955
edit: I should add a quick explanation for the circuit too.
You've essentially just got 2 strings of LEDs, each string is hooked up to 1 output on your PIC. This'd be fine for normal power LEDs, but since you need it to be visible in daylight we'll need LEDs with higher power
The PIC can only supply 20mA of current from each output, which is pretty small. So instead of using the PIC to power the LEDs, we'll use the PIC to power a MOSFET
A MOSFET is basically an electrical switch that can handle lots of power going through it. It has 3 pins, Gate, Drain, and Source.
You connect one side of your circuit to the "Source" pin, and the other side to the "Drain" pin. The "Gate" pin is what turns it off/on.
In our case, we connect the picaxe output to the "Gate" pin - when we set the pin "low" the switch is off, when we set the pin "high" then the gate will turn on and allow current to flow from Drain to Source, completing our circuit and turning your LEDs on (yay!)
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