Turkey's Prime Minister Ahmet Davutoglu was in London last week, telling the Western media how helpful Ankara was being in the struggle against the terrorist ''Islamic State'' in northern Syria and Iraq.
Turkey is doing everything it can, he said although, of course, ''We cannot put troops everywhere on the border.''
Turkey's open border has become a sore point with its Western allies, who suspect that President Recep Tayyip Erdogan is deliberately allowing a steady flow of recruits and supplies to ''Islamic State'' because he still wants the Sunni rebels, most of whom are jihadi extremists, to overthrow Bashar al-Assad, Syria's Shia ruler.
(Erdogan is no jihadi, but he is a devout and militant Sunni Islamist.)
But Erdogan's motives are irrelevant, because Turkey simply cannot put troops everywhere on its 820km border with Syria.
Or so says Ahmet Davutoglu, and only an enemy of Turkey (or somebody with a grasp of basic mathematics) would say otherwise.
I am no enemy of Turkey, but I can do basic arithmetic.
If you stationed Turkish troops along the entire length of the Syrian border at 10m intervals - that's enough for a machine-gun nest every 50m - it would take about 82,000 soldiers to cover the entire 820km.
The strength of the Turkish army (never mind the navy and air force) is 315,000 soldiers.
Maybe Turkey doesn't have that many machine-guns, but it's not a poor country, and machine-guns are quite cheap on the international market.
Or maybe it would prefer to use some other equipment instead: a good fence and some motion-detectors would help.
But the main requirement is manpower, and not very highly skilled manpower at that.
The Turkish army has a few other jobs, but not any high-priority ones.
Even if you allow for frequent rotation of the soldiers manning the border, it would take much less than half the strength of the Turkish army to shut the border to foreign fighters.
Maybe a few jihadis would still get through, but the vast majority wouldn't.
The only reason Ankara doesn't shut the border is that it doesn't really want to.
Cutting off the flow of jihadi volunteers to Syria would not greatly change the local military balance: IS uses them mostly as mere cannon-fodder.
The point is that Turkey is not fully committed to the destruction of Islamic State, and indeed will give IS deniable help in order to further the goal of a Sunni victory in Syria, despite being part a ''coalition of the willing'' that is nominally dedicated to destroying IS.
The same goes for Saudi Arabia, although it has sent some token aircraft to bomb IS.
Riyadh tries to prevent any Saudi citizens from going to fight for IS, and it certainly does not want the IS brand of radicalism to come to the kingdom.
Indeed, Saudi Arabia has already started building a 900km high-tech wall along its border with Iraq to stop IS activists from entering the country.
But it is not a long way from the Wahhabi brand of Sunni Islam that is promoted by Saudi Arabia to the ''takfiri-salafist'' doctrines espoused by the IS militants.
Saudi private individuals have been a major source of financing for IS and until recently Riyadh just turned a blind eye to it.
Even now, Saudi Arabia doesn't want Islamic State destroyed if that means Assad gets to stay in power in Syria.
Then there's Iran. In Iraq, where Islamic State controls half the country's territory and threatens a Shia-dominated regime, Iran and the United States are fighting almost side-by-side to defend Prime Minister Haidar al-Abadi's Government.
(They don't actually talk to each other, but they each tell the Iraqis where they are planning to bomb so there are no collisions over the target areas.)
But next door, in Syria, it's different. Iran has sent troops, weapons and money to defend Bashar al-Assad's regime, while the United States is still pledged to overthrow it.
They both see Islamic State (which controls about a third of Syria's territory) as an enemy, but Washington still believes that it can create some other, more ''moderate'' army of Sunni rebels that will eventually take Assad down.
And Russia, of course, still supplies Assad with weapons, money and diplomatic support - but despite its own difficulties with jihadi rebels back home in the North Caucasus, Moscow is not participating in the military campaign against Islamic State.
Its quarrel with the United States over Ukraine is too fierce to permit that degree of co-operation elsewhere.
And so on, and so forth. Not one of the major outside powers that is opposed to Islamic State in principle has a clear strategy for fighting it, nor are they willing to co-operate with one another.
So IS will survive, at least for some years to come, despite the horrors it inflicts on the innocent people under its control.
It may even expand a bit more, though the end of the siege of Kobane shows that it is far from unstoppable.
• Gwynne Dyer is an independent London journalist.
