Right, a hypothetical day.
A river you have never fished before, typical medium sized spate stream.
It holds wild browns, grayling, and a few stocked brownies.
Mid morning, nothing rising, water in good order.

What method and flies would you start with?????????

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It is in truth not glory,nor riches, nor honours that we are fighting,but for freedom - for that alone, which no honest man gives up but with his life.(Declaration of Arbroath, 1320)

Black beadhead woolly bugg3r, cast tight to the bank and swim it across. If you're allowed to use more than one fly, I'd add a Brown Trout Streamer. The WB in particular is, IMO, the best "nothing else going on" prospecting rig in fly fishing.

Then, assuming it's summer, I'd start in with the terrestrial action, either throw some hoppers or ants. Trout may not be rising, but if they see a hopper float over them, the WILL, more likely than not, be rising to nail it. It's like freaking crisps to them, you can't eat just one.

If that doesn't work, I'd start in on the spiders and wetflies. In the earlier spring and later fall, I'd start with the spiders and wetflies and then go to the WB and streamer action.

Grouse

Size 14 sparsely-dressed GRHE, cast under the bank/tree cover and fished upstream.

Search out as much water as possible with the one fly.

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Easy, fish the duo, size 12 klinkhammer on a dropper which will cover the surface option and a nymph under the klink size 14-16. Fishing this way covers both options. Very successful method for searching water quickly.

Walker

At this time of the year..........or tommorow ill start with a scruffily tied goldhead grhe on point, b&p spider on the dropper and maybe a mallard n clarett in the middle. If that doesnt do it change the b&p for a reg tag on top dropper and the grhe for a goldhead pheasent tail. if it was a bit sunnier( dont know why i always prefer to fish dries when its sunny)and i wanted to fish dries id wander upstream flicking various parachute hackled flies in front of me.
Tommorow if all that fails ill find some deep holes n chuck a peeping caddis into them until it works.