The phrase six of one and half a dozen of the other denotes a situation in which the alternatives are considered equivalent—synonym: (as) broad as long.
This phrase is first recorded, as six of the one and half a dozen of the other, in the journal of Ralph Clark (died 1794), a British naval officer who was first lieutenant on HMS Sirius when the ship was wrecked in March 1790 on a reef at Norfolk Island, in the Pacific Ocean, off the east coast of Australia (HMS Sirius was the flagship of the First Fleet, which set out from England in 1787 to establish the first European colony in New South Wales, Australia); this is what Ralph Clark wrote on Saturday 24th April 1790 when the crew and the convicts were stranded on Norfolk Island:
a great Surf on the Reef —— Captain Hunter and Johnstone Lieuts Creswell and Johnstone went to Mount Pit to day —— Majr. Ross discoverd that Captain Hunters Servants (and he beleves his own Servents has Some hand it) have drunk or made away with all his Rum which he has been able to Save out of the Sirius —— of all the places in the World this is the greatest nest for Rascals it is impossible to trust any one of our men hardly much more any of the Convicts in Short there is no difference between Soldier Sailor or Convicts there Six of the one and half a Dozen of the other —— old Elliock was a man Majr. Ross placed the greatest confidence in and he and Ancott have Repaid the Major for the Confidence he placed in them as all Rascals Repay good Masters —— he has turnd them both of as has Captain Hunter his black Villian*——
—from The Journal and Letters of Lt. Ralph Clark 1787-1792 (University of Sydney Library, 2003)
(* Clark seems to be alluding to the execution of the rum thieves.)
The second-earliest instance of the phrase, also as six of the one and half a dozen of the other, is from The Edinburgh Review, or Critical Journal (Edinburgh) of November 1820, in the review of The Bakerian Lecture. On the Composition and Analysis of the Inflammable Gaseous Compounds resulting from the Destructive Distillation of Coal and Oil; with some Remarks on their relative Heating and Illuminating Powers, by the English chemist William Thomas Brande (1788-1866), published in the Philosophical Transactions for 1820:
We must also take leave to observe, that in some parts of his inquiry, Mr Brande’s train of reasoning is a little fallacious. It is very like what mathematicians call, arguing in a circle. For example; he conjectures, from the specific gravity of oil gas, that it is composed of one volume of hydrogen and three of olefiant. He then makes a mixture in this proportion, and exposes it to the action of heat till the olefiant gas is decomposed—and he finds, of course, that it has increased by the quantity of olefiant gas in the original mixture—which only proves, that the olefiant gas has changed, by decomposition, into double the quantity of hydrogen;—which he knew beforehand must be the case,—and that the remainder is the hydrogen which he himself put into the mixture, and the quantity of which he might, therefore, have guessed without this process. He goes on to expose the mixture with chlorine to the action of light—and finds, after the chlorine and olefiant gas have been absorbed, that the original quantity of hydrogen remains in the jar—thus demonstrating that there are six of the one and half a dozen of the other.