Snapshots of the Week That Was from Bloomstan.

On Wednesday, the mayor and five other rich New Yorkers each chipped in $250,000 of private money to cover the $1.5 million cost of restoring state Regents exams in January.

On Thursday, he and George Soros each chipped in $30 million, and taxpayers will pay $67 million, for programs to help ex-convicts get jobs and keep minority kids in schools.

Then he dumped a deputy mayor he had brought in to breathe fresh life into his third term.

So it goes in Year 10 of the Bloomberg Way, with the week’s events illustrating the growing weirdness of his mayoralty. He is passionate about improving the lives of the poorest New Yorkers and is eager to spend his own money to do it, and yours, but his attention to core city issues remains maddeningly erratic.

Most telling, his willingness to use his wealth as an instrument of public policy signals he is beginning to think about the transition from City Hall to full-time philanthropist when his term expires in 2013. He clearly intends to be a player who will cast a shadow over his successor.

The combination of government power and private wealth, estimated at more than $20 billion, makes Bloomy unique in New York history. He’s the richest man ever to hold elective office in the United States.

Except during his campaigns, where he spent about $280 million in reported expenses, Bloomberg mostly had been circumspect about mixing his money with the public’s.

For several years, he funneled millions as “anonymous” through the Carnegie Corp. to arts and cultural groups. It didn’t hurt that he could count on their support for such things as bucking the term-limits law.

More recently, the lines separating his job, his media company and his family foundation are becoming muddy. A former deputy mayor, Dan Doctoroff, is now president of Bloomberg LP, while his first deputy mayor, Patti Harris, doubles as head of his foundation. Some city aides are also on his personal payroll, giving double dipping a new meaning.

Harris, other City Hall aides and people from Bloomberg LP and the foundation are known to meet in the foundation’s $45 million mansion on East 78th Street. Whether they’re doing public business isn’t clear. The entities have all sort of merged in a way that is convenient for Bloomberg and his team, despite questions about whether the mix is kosher.

The result for taxpayers is clear; Bloomberg’s divided attention is the reason his third term still has not found its groove. Deputy Mayor Stephen Goldsmith was canned last week after only 14 months, far longer than Cathie Black’s cup of coffee as chancellor, but still a sign of failure.

Goldsmith was supposed to think big thoughts about reforming government, but also got the job of deputy mayor for operations, which supervises the delivery of key services. Traditionally, that job goes to the first deputy mayor, but Harris didn’t want it.

Goldsmith was out of town, as were the mayor and Harris, during the Christmas blizzard, but Goldsmith was the fall guy. So now he’s gone.

“It’s very unfair,” said one insider. “They gave him the wrong job and, of course, he failed.”

Finally, there was another key event last week, and it might be the most revealing about the Bloomberg Way. A devastating report on the city budget revealed the impact of the unaffordable spending on his watch.

The Citizens Budget Commission says the combined annual cost of pensions, retiree health care and debt service will more than double in the decade that ends in 2015. The city spent $9.2 billion in 2005 on those items, but the tab will be a staggering $18.7 billion in three years. They will consume 25 percent of the budget and squeeze everything else.

The group calls these three unsustainable obligations “legacy costs.” The irony is rich. Bloomberg is the king of bucks, but unless he finds a way to reduce these costs, it’s his legacy they will destroy.

‘Nonprofit and loss’ statement

Aiming to cut big pay packages at nonprofit groups getting state dough, Gov. Cuomo has put his finger on a key issue — but at the wrong end.

The proper role for him is to stop New York’s willy-nilly funding of so many questionable groups. Eliminating legislative slush funds and pork-barrel spending would save hundreds of millions of dollars and end the corrupt practice of lawmakers buying political support with taxpayer grants to nonprofits, some of which are just patronage dumps.

Instead, by focusing on salaries, Cuomo could hurt the most successful organizations, which must compete for top leaders. While they get some state funding, many perform important civic functions and raise much of their budgets through private donations and fees to users.

The problem of corruption is in Albany, and that’s where the scrutiny should begin.

‘Forever’ in Times’ debt

The New York Times editorial page has lost the ability to shock, but still manages to amaze. Its latest blast from outer space is a call to eliminate the nation’s debt limit. Not, to be clear, the crippling debt itself — just the cap on it.

The goal is to prevent Republicans from ever again using the cap to negotiate spending cuts. So as the paper sees it, if there’s no debt limit, there can be no fights over spending.

Brilliant! Of course, without a cap, there will be endless debt and additional credit downgrades. But that’s fine with the Times, as long as Democrats are doing the splurging.

W. House has its Irish up

Eagle-eyed readers, this scoop is for you. A photograph published with Wednesday’s column showed President Obama and Vice President Joe Biden in a meeting on the debt-ceiling talks. Above and behind Obama’s head was a sign that said, “Help Wanted,” and, in letters too tiny for my eyes, “No Irish Need Apply.”

Reader Scott Vankuren, among others, was able to read the offensive sign and wondered what the hell it’s doing in the White House. “With a great many being of Irish descent, I wonder if you take umbrage in the president’s choice of office decor, much the same as if your media outlets posted signs for white and black water fountains, or signs ordering blacks to the rear of the bus.”

Here’s the story. The photo, taken by a White House photographer, captured the president in the office of his chief of staff, Bill Daley, whose name declares his Irish ancestry.

A White House spokesman adds the details: “The sign, an original dating from 1915, was a gift to the chief of staff decades ago. Mr. Daley has displayed the sign in every office he’s ever had as a reminder of a time when discrimination against Irish immigrants was widespread. The sign serves as a constant reminder of the need to remain vigilant against discrimination and intolerance in any form.”

Now that’s a happy ending. Case closed.