The humorous phrase elementary, my dear Watson, and its variants, mean that the solution to a problem is very straightforward and easy.—Cf. the phrase easy-peasy.

A variant of elementary, my dear Watson occurred, for example, in Ah, but do you know the provenance?, by Hanaba Welch, published in the Standard-Times (San Angelo, Texas, USA) of Sunday 14^th January 2024 [page 10A, column 2]:

If my detective persona is enjoying itself, maybe it’s because I’m inspired by a Sir Arthur Conan Doyle spin-off set in Texas: “A Study in Crimson: Or the Tale of the Red Man” by Will Brandon.
Yep, it’s elementary my dear reader. When provenance is the goal, don your Sherlock hat and let your imagination string together some believable possibilities, the wilder the better.

The phrase elementary, my dear Watson refers to two fictitious characters created by the British author Arthur Conan Doyle (1859-1930): the British detective Sherlock Holmes, and his friend and associate, the British physician Dr. John H. Watson.

However, this phrase is not (in this form at least) found in any of Arthur Conan Doyle’s stories.

The earliest occurrences of the humorous phrase elementary, my dear Watson, and of its variants, that I have found are as follows, in chronological order:

1-: From the Northampton Mercury (Northampton, Northamptonshire, England) of Friday 15^th November 1901 [Vol. 182, No. 9,438, page 6, column 3]:
—Note: several British newspapers reprinted this text from December 1901 to May 1902:

SHERLOCK HOLMES’S LATEST!

One winter’s morning, a few years after my marriage, I was lying by my hearth, smoking a red herring, and nodding over the Encyclopedia Britannica, for my day’s work had been a hard one. Since first meeting Shylock Combs my practice as a doctor had, as a matter of course, rapidly declined. My presentation clock had chimed 4.47 a.m., and I was in the act of blowing out the gas when I heard the clang of my front door speaking tube. Thinking it must be the milkman I went into the hall, opened the door, and, to my astonishment, Shylock Combs stood upon my step. “Ah, Potson,” he said; “I hoped I should not be too early to catch you. I perceive the wind has changed round to N.N.E. by S.W. again.” I was astounded, as he had not had time to observe the thermometer in my bedroom. He noticed my amazement and smiled that wonderful smile of his. “Elementary, my dear Potson,” he said; “I observed the left-hand side of your moustache inclined about 47⅝ degrees towards the west, and coming as I did from Butcher-street I at once deduced from which quarter the wind was blowing.” I was about to make one of my usual remarks when he placed his fingers on his lips and strode into my sitting-room. Naturally, I followed. He threw himself into my easy chair, took his corncob pipe out of his pocket, and filling it with snuff, smoked for some hours in silence. “Potson,” at length he said, “Can you spare a few weeks of your valuable time to assist me in unravelling a most interesting little problem?” I replied that I thought I could, as it was my busy time, and I knew my friend Dr. Phillipson would look after my one patient during my absence. “Very good,” he replied, “meet me at Waterloo Station at 10.30 in the morning, and do not forget to bring your Gatling gun with you.” He rose from my chair, and as he was leaving the room said to me, “I am pleased to see that you have recovered from your recent indisposition. You are looking quite yourself again.” I was completely taken aback, as I had not seen my friend for at least eight months. “Why, however did you know?” I commenced. Once more that curious smile passed over his face. “Very simple,” he replied, putting on my best silk hat. “I observed a box of Charles Forde’s Bile Beans for Biliousness on your sideboard as I was leaving. Good morning.”

2-: From the column By the Way, published in The Globe and Traveller (London, England) of Thursday 27^th February 1908 [No. 35,168, page 1, column 4]:

“I was in hospital,” said a witness at the Clerkenwell County Court, “but by a mistake the wrong card was attached to my bed, and I was treated for what the man in the next bed was suffering from.” These little slips will happen. Your really tactful man passes them off with a smile.
At the same court there was a discussion as to the means of a certain debtor. “How much do you spend on cigarettes a week?” asked the lynx-eyed counsel. “A man who comes here with his fingers saturated with nicotine stains must spend a considerable amount of money on cigarettes.” (“My dear Holmes,” I cried, astonished, “how——?” “Elementary, my dear Watson,” he replied carelessly).

3-: From the column By the Way, published in The Globe and Traveller (London, England) of Friday 17^th July 1908 [No. 35,288, page 1, column 4]:

“My hat is more than five years old,” said a woman debtor at Whitechapel. Judge Bacon leaned back in his chair, placed his finger-tips together, closed his eyes, and smiled a quiet, tight-lipped smile. “Women did not wear hats like that five years ago,” he said, “I observe and I know.” (“My dear Bacon, how——!” “Elementary, my dear fellow, quite elementary!”)

4-: From the column By the Way, published in The Globe and Traveller (London, England) of Friday 21^st August 1908 [No. 35,318, page 1, column 4]:

From an article on Empty London.—“The clubman taps the kerb, wondering where to go, and that sudden jerk of the head shows that he has made up his mind to dine on the Scottish Express to-night.” (“My dear Holmes, how?” “Elementary, my dear fellow, quite elementary”).

5-: From the column Voice of the People, published in The Times-Dispatch (Richmond, Virginia, USA) of Tuesday 24^th August 1909 [No. 17,992, page 6, column 7]:

SIGNALING TO MARS.
An Elementary Problem, Says Professor Pickering, of Harvard.

The possibility of signaling to the planet Mars is merely a question of elementary mathematics. At least, that is what Professor Pickering, of Harvard, says, and he expresses surprise that his plan of utilizing mirrors for the purpose of shunting the rays of our sun onto our nextdoor neighbor should have excited such widespread interest and discussion. It is such a simple little problem that any one should be able to take a pad and pencil and work it out in ten minutes. “Elementary, my dear Watson,” as Sherlock Holmes was wont to say. “Elementary.”

6-: From Langford has not gone back, as some believe, published in The Buffalo Commercial (Buffalo, New York, USA) of Monday 18^th April 1910 [page 7, column 5]:

When Sam Langford knocked out Jim Barry the other night in Los Angeles nothing in particular was proven. […]
[…] Langford’s superiority over second class middleweights—and even heavyweights—is a matter of course. Other fighters acknowledge him to be one of the greatest men in the ring today. That part is elementary, my dear Watson.

7-: From the column Notes of the Day, published in The Globe and Traveller (London, England) of Thursday 23^rd June 1910 [No. 35,887, page 1, column 3]:

IN ERROR.

“Uncle, do you know the difference between an apple and an elephant?” “The difference between an apple and an elephant, my boy? Let me see. No. No, I don’t know the difference between an apple and an elephant.” “You’d be a smart chap to send out buying apples, wouldn’t you, uncle?” We had always imagined this ancient dialogue to be fictitious, but recent events which have come to light, show us that we were wrong. The “uncle” of the dialogue is an actual living man. You want to know some details about him? Very well. He is an enthusiastic, but distinctly erratic, sportsman, and quite recently he has been travelling in Uganda. How do we know? Elementary, my dear Watson, elementary. When we saw in the papers that four bull-elephants and two cow-elephants had been shot “in error” in Uganda, we knew that that was uncle. We defy any ordinary man, possessed of the happy gift of knowing an elephant from an apple, to shoot one of the former species “in error.” He might shoot him in anger, in contempt, or through pure animal spirits, but he would not shoot him in error. No, it was uncle right enough, bless his little old heart. We can see the dear old gentleman at it. He is walking meditatively along, when a bull-elephant saunters into view. He starts. This is something new to him. “There are more things in Heaven and Earth, Horatio,” * he mutters, as he loads his gun. Then, as the elephant comes nearer, his face clears. “Silly of me,” he chuckles. “Of course I know what it is now. It’s an apple.” Bang! And up jumps the death-rate among Uganda elephants.

* This is a quotation from The Tragedy of Hamlet, Prince of Denmark, by the English playwright and poet William Shakespeare (1564-1616).

8-: From the column Editorial Jottings, published in The Sun (Baltimore, Maryland, USA) of Saturday 6^th August 1910 [Vol. 147, No. 82, page 6, column 3]:
—Note: I have not found this text in The Kansas City Post (Kansas City, Missouri, USA):

A careful pursuit of the Crippen case fully vindicates the opinion Sherlock Holmes had of the sagacity of Scotland Yard. “Elementary, my dear Watson, elementary.”—Kansas City Post.

9-: From The Washington Herald (Washington, District of Columbia, USA) of Wednesday 10^th August 1910 [No. 1,404, page 4, column 2]:

A Real Sleuth.

The notion that Sir Arthur Conan Doyle’s interesting creation, Mr. Sherlock Holmes, is overdrawn, or not to be paralleled in real life, is erroneous. Mr. Holmes was an astute genius; but there are living, breathing de-tec-i-tiffs who measure up quite as grandly to the demands of various occasion as ever Mr. Holmes did.
Mr. Holmes always insisted that his methods were simplicity run riot. His most famous deductions invariably were characterized by himself as “elementary, my dear Watson; elementary.” Mr. Holmes merely put two and two together; never once did the putting together of two and two fail to make four. The problem was to locate the several twos. This Mr. Holmes persistently did with great neatness and dispatch. Usually, it was no trouble whatever to the incisive Sherlock. Everything was explained eventually to everybody’s complete and lasting satisfaction—that is, everybody’s excepting Lestrade’s, of Scotland Yard. That picturesque and somewhat stupid person generally was forced to retire “thoroughly crestfallen.”

10-: From the column By the Way, published in The Globe and Traveller (London, England) of Saturday 12^th November 1910 [No. 36,009, page 1, column 4]:

A Welsh gardener, we read, claims to be able to cure certain ailments by the laying-on of hands. “He attributes this to the fact that he is the great-grandson of a man who had shot and eaten a bit of an eagle.” (“My dear Holmes, how——!” “Elementary, my dear fellow, quite elementary.”)