There's nothing in the original specs that is either too unusual for a stopwatch/timer or hard to do.

I would start with a 10Hz clock source (oscillator or oscillator/divider chain) and follow it with four stages of the old standby counter series - 74LS192/74LS75/74LS47/LED. Start/stop is a switch in series with the clock (debounce it if you are afraid of noise). Reset is a button tied to the reset line of all counters. The only trick is the coupling between stages; there would be two sets of connections, one for counting up, and one for counting down. They would be gated by both the up/down select line and the correct decoding of the outputs to trigger the next stage - carry/borrow out will come in handy here.

/mike

 

One solution,,, you're going to need to use cd4510 bcd counter, cd4511 bcd to seven seg driver, and a 10hz clock source, say a 555 timer or cd4093 osc, and logic gates to send the reset pulse to the last counter..

I'd go this route too. The only part that requires much effort is the tens second counter. The CD4510 datasheet has parallel and ripple clocking examples.