One day, a famous government official was passing along a road. He saw an old monk teaching Buddhism. Hardly unusual except that the monk was seated on a tree branch.
The official asked the elderly monk what he was doing. After all, the monk was in a very precarious position. One wrong move and he could fall to his death!
The monk replied that the official’s position was even more precarious. If the monk made a careless move, he alone might be killed. But if the official made a mistake, it could cost the lives of thousands.
The official considered this and decided that it was a very good reply. He told the monk that if he could explain the essence of Buddhism in one sentence, he would become the monk’s student.
“Easy!” said the monk. “The essence of Buddhism is to avoid all that is evil, to embrace all that is good, and to purify one’s mind.”
The official scoffed, “Is that all? Even a child of three knows that!“
The monk replied that while it was true that a child of three may know it, there was no certainty that a man of eighty could practice it.
a buddhist perspective - abuddhistperspective - Heart of the Teachings
Suffering and the cessation of suffering?
with metta
The sentence is part of a twofold summary of the way:
To avoid all evil, to cultivate good, and to cleanse one’s mind—this is the teaching of the Buddhas.
Enduring patience is the highest austerity. “Nibbana is supreme,” say the Buddhas. He is not a true monk who harms another, nor a true renunciate who oppresses others.
Not despising, not harming, restraint according to the code of monastic discipline, moderation in food, dwelling in solitude, devotion to meditation—this is the teaching of the Buddhas.
It is found in the Dhammapada and DN14.
It is usually known and recited as Ovada Patimokkha Gathā and is associated with the Magha Puja, which is a relatively important date in Theravada Buddhism, second only to Vesākha Puja.
I recall hearing the gathā being recited at the end of the fortnightly recitation of the Patimokkha by Thai forest bhikkhus of Dhammayut order/fraternity, don’t know if this is done by all from other orders/fraternities.
Khantī paramaṃ tapo tītikkhā
Nibbānaṃ paramaṃ vadanti buddhā,
Na hi pabbajito parūpaghātī
Samaṇo hoti paraṃ viheṭhayanto
Patient forbearance is the foremost austerity.
Liberation is foremost: that’s what the Buddhas say.
He is no monk who injures another;
nor a contemplative, he who mistreats another.
Sabba-pāpassa akaraṇaṃ,
Kusalassūpasampadā,
Sacitta-pariyodapanaṃ:
Etaṃ buddhāna-sāsanaṃ.
The non-doing of any evil,
The performance of what’s skillful,
The cleansing of one’s own mind:
This is the Buddhas’ teaching.
Anūpavādo anūpaghāto
Pāṭimokkhe ca saṃvaro
Mattaññutā ca bhattasmiṃ
Pantañca sayan’āsanaṃ.
Adhicitte ca āyogo:
Etaṃ buddhāna-sāsananti.
Not disparaging, not injuring,
Restraint in line with the monastic code,
Moderation in food,
Dwelling in seclusion,
Commitment to the heightened mind:
This is the Buddhas’ teaching.
When this is, that is.
From the arising of this comes the arising of that.
When this isn’t, that isn’t.
From the cessation of this comes the cessation of that.
-so it isn’t one sentence. 
with metta
@Mat, worry not, I think the point of the topic is not to affirm a single sentence as the summary of the teaching.
It is an opportunity to acknowledge that to some small summaries like those may well be all that they can carry and may be as well just enough to make them feel better about things, and who knows, point them towards the end of suffering.
Heedfulness
Everything is impermanent 
Processes and conditioning is all there is .
SN 56.11, first discourse
…tam kho panidam dukkham ariya saccam, parii neyyam
…tam kho panidam dukkha samudayam ariya saccam, pahaatabbam
…tam kho panidam dukkha nirodham ariya saccam, sacci katabbam
…tam kho panidam dukkha nirodha gaaminii patipadaa ariya saccam, bhavetabbam
I think Ud 1.2 will be my submission:
This not being, that is not;
from the cessation of this, that ceases.
Here is my submission for “Buddhism Explained in One Sentence,” granted it’s a complicated compound sentence. 
“Pain and suffering arise from two sources - (1) our unskilful craving, and (2) our attachment to the false narrative that creates and supports our false sense of self (i.e., our ego); freedom and contentment are experienced by freeing ourselves from the views and behaviours that perpetuate unskilful craving and ego-clinging.”
or perhaps, Buddhism can be explained in just the last part of my submission:
“Freedom and contentment are experienced by freeing ourselves from the views and behaviours that perpetuate unskilful craving and ego-clinging.”
The Buddha said “look this way and you will discover the way out is the way in”.