Just roll a Toblerone.

there is such a thing as a 3-sided dice here's a link!:https://www.google.com/search?q=3 sided die&rlz=1CAZICU_enUS971&source=lnms&tbm=isch&sa=X&ved=2ahUKEwjJ_MHj-LH3AhX2s4QIHWHxBTgQ_AUoAXoECAEQAw&biw=1366&bih=617&dpr=1&surl=1&safe=active&ssui=on

Can only spell casters use this or can a rogue use it to check his work?

Jesus Christ people, just roll 1d6.
1-2 = 1
3-4 = 2
5-6 = 3 

It's not that hard.

So if this wand just uses an action, is it reasonable to assume that its casting is silent? Since it's not specified that a command word is spoken?

lol

Per the DMG a command word is the default, so unless specified, there is one.
https://www.dndbeyond.com/sources/dmg/treasure#ActivatinganItem

I've never liked this item, because either you waste the wand in rooms that have no secrets, or you need a DM willing to tell you that there mayyyyy be a secret door or trap nearby, but you're not sure. Usually your perception either spots it, or you miss it.

I always home-rule that once activated, the wand stays active for an hour, that way it can get more use when in the appropriate space.

flip a coin and add one?

Quote from Pancakeli >>

Quote from SeeboMurnig >>

Can this be used as an Arcane Focus as well?  Sorry if this is a dumb question.

 No worries it's not a dumb question, and yes it's a wand still, so it can still be used as an arcane focus. It just also has a special ability on it. Similar to how an enchanted longsword can still be used as a longsword

dumb

What does a d3 even look like? 

What?

No. Roll a d6.

1=1
2=2
3=3
4=1
5=2
6=3

Why would you do it any other way?

hi

What does a d3 even look like?