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Like most programming problems, if you understand the logic behind what you’re trying to accomplish, all you need to do is figure out a way of representing that logic with programming code. So let’s think for a moment about the problem we have to solve.
How would the meaning of the sentence change if figure out was changed to to figure out?
There is no difference in meaning between to-infinitives and bare infinitives. The use of one form or the other is generally determined by the verb controlling the infinitive clause.
Typical examples of a verb requiring the use of a to-infinitive are:
I want to go home.
I need to go home.
Other verbs admit both to-infinitives or bare infinitives, e.g.:
I will help you go home.
I will help you to go home.
Auxiliary and most modal verbs don't admit a to-infinitive, e.g.:
I did go home.
I will go home.
I should go home.
I needn't say anything.
I need go home. (rare)
A few modal verbs demand a to-infinitive, e.g.:
I was unable to go home.
I ought to go home.
The lexical verb need to and the modal verb need/needn't are an interesting case. Note that whereas need/needn't doesn't admit a to-infinitive, need to demands it. As a consequence, it is possible to find both:
I need to go home. (common)
I need go home. (rare)
A search in Google's n-gram viewer reveals that the use of need + bare-infinitive is rare.
The example in this question:
[...] all you need to do is figure out a way [...]
is compounded by the use of two verbs, need that usually takes a to-infinitive and do that takes a bare infinitive.
Based on the above arguments, one would expect the following uses:
all you need is to figure out a way
all you need to do is figure out a way
However, another search in Google's n-gram viewer reveals that the following alternative, although rare, is also possible:
all you need to do is to figure out a way
Both forms are correct, and they mean the same thing. The form without the "to" is more common. (I did a search in the COCA corpus. The form without the "to" is six times more common.)
Actually, using to in front of figure out would be correct and acceptable, but unnecessary, since figure out here is governed by the first to. In a sentence without the first to, you would have to supply it:
All you need is to figure out...
Since it would be incorrect to say: All you need is figure out, we know that to is not optional in your sentence - it is required, and present, supplied by the first to. However it can be restated with no change in meaning.
As Dangph says, both forms mean the say thing. I would say these represent stylistic options. Using "to figure out" gives the sentence a more formal and precise tone to it. However, the beginning of the next sentence "So let's think..." makes reveals that the tone of the paragraph is meant to sound informal.
