Another subject-verb agreement [duplicate]

Solution 1:

The most common use of need is as a regular catenative verb, taking a to-infinitive as its complement. In that use, it inflects normally (need/needs/needing/needed), can follow an auxiliary verb, and so on: "I might need to talk to him."

However, in a rather formal style of English, there also exists an auxiliary verb ("helping verb") need, which is followed by a bare infinitive, and is only used in negative polarity contexts: with not, with only, in questions, and so on. Additionally, it only exists in the present tense. Aside from these two restrictions, it's very much like other auxiliary verbs (can, may, should, etc.): "We need not discuss it."

So your first example ("[…] one need only look […]") is correct — using the auxiliary verb need — whereas your second example ("[…] one needs only look […]") is wrong — it should either be "one need only look" (using the auxiliary verb), or "one only needs to look" (using the regular verb).

Another verb that behaves this way, by the way, is dare.