I often encounter a blind allin shove (live tables), both in MTT and cash games (mostly by degenerate gamblers, and I encounter them every day on the table)
Assuming:
1, The all-in maniac did not look at his card when shoved
2, no one else call or raised, action is back to hero whom is last to act (heads up).

What hand range does the hero need to call with
1, 60% equity
2, 50% equity
3, 40% equity (rare situation in tournaments)

Other reference:
From ultimate Texas holdem (a casino table game)
https://discountgambling.net/ultimate-texas-holdem/
The hands that are good enough to raise 4x:
pair of 3’s or higher
Ace + anything
any suited King, or King and 5 or higher
Queen and 6 suited, or Queen and 8 or higher
Jack and 8 suited, or Jack and Ten

Since one of your stipulations is that you are last to act, we can assume you did raise and villain shoved. (unless you are BB)

That should mean your hand is stronger than any two and you call every time. Except for some weird BU/SB spots where you raise ultra wide, but you probably shouldn’t do that if there’s a decent chance your opponent will shove on you.

This sounds like a perfect job for Equilab. If you don't yet have this tool you can get it for free at pokerstrategy.com. You can check your range equity against any other range (or ranges if multiway), including a random hand. It's a great tool for off the table study to answer questions just like the one you asked.

I will assume this is pre-flop and the villain open-shove amount is Xbb . Pot depends on positions so to make things simple, assume both hero and villain are not in the blinds. In bb units:

Pot = X + 1.5
EV = eq*(2X + 1.5) – X
For break-even, eq = X/(2X+1.5)

This says the minimum equity for break-even is less than 50%. For example, if the shove is 3bb, the needed equity is 3/7.5 = 40%. For a 10bb shove, equity of at least 46.5% is needed.

What should your calling range be? Well, if villain did not look at his cards, you can assume he holds any two cards, so his range is 100%.

From Equilab, for the 3bb example, you have at least 40% equity vs. a 100% range with the following calling range is

22+, A2s+, K2s+, Q2s+, J2s+, T2s+, 92s+, 82s+, 73s+, 64s+, 54s
A2o+, K2o+, Q2o+, J2o+, T2o+, 93o+, 85o+, 75o+

This is the range so that every holding has at least 40% equity versus ATC.

For the 10bb shove with a 46.5% equity requirement, the calling range is

22+, A2s+, K2s+, Q2s+, J2s+, T4s+, 96s+, 87s
A2o+, K2o+, Q2o+, J5o+, T7o+, 98o

^You probably don't want to call hands barely making the pot odds cutoff if you're at a table with guys going allin blind.

@starmahal: looks good but that’s not what OP was asking. You’re calculating ranges with both players not in the blinds and villain open-shoving, which is not possible in a scenario where hero closes the action and action is *back* on him.

Things get significantly more complicated with players left to act after hero calls villains shove.

Right, I realized that I may not be addressing all the details but I tried to focus on the main issue of dealing with a maniac shove.

When making these calculations, allow for rake in cash games. When your edge is 0.5%, then rake can obliterate it.

This

I spent a few hours farting around with equilab a while back when I wanted to answer this kind of question. You end up calling with a surprisingly wide range.

Most time I would flat call first if I had a hand that I am willing to go allin with (assuming no one call or raised before me), villain shoved allin (villain will shove allin everyhand without looking at his card, most time declared allin before his hand was dealt, so it is a out of turn move), then the action fold back to me. If anyone else involved in the hands, my range would tighten up like a rock player's allin range.

Every single hand you think is good enough to limp should call a shove from any 2.