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Make:Electronics Experiment 15 (break to make transistor circuit)

Hi there, I'm having problems with the early part of the make to break circuit, the schematic is Figure 3-89:

I'm using components from Radioshack kit specifically assembled to go with the book. The strange thing about my PN2222A transistor is, if I use a 10K resistor as shown in the schematic, there's not enough current going to the base of the transistor (when the switches are open) to activate the relay.

I found that only using a resistor of 100ohms in place of that, works. I tried anything else as low as 200ohms, and they all don't work.

However, this leads to two problems:
(1) This means I need to use something much lower than 100ohms in the lower resistor (1K in the schematic). I don't have a smaller resistor than that, so I simply substituted a jumper cable instead. But I believe this will burn out my 12V AC adaptor quickly?

(2) When the relay is being activated (switches open), the 100ohm resistor gets hot very quickly and I see smoke coming out of it.

What are possible alternatives?
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