When Joe Torre managed the Yankees, he sometimes liked to jot down possible batting orders on a piece of paper. His scribbling didn’t necessarily have to be on a lineup card; it might be on a napkin or the back of an envelope or a notepad. Like a calculus student working through an equation, Torre just wanted to see what it looked like -- how it fit together in balance, in strengths and weaknesses.

Alex Cora and Aaron Boone have probably been going through the same exercise the past couple of months, as they prepare to take over as managers of the Red Sox and Yankees, respectively, and the same could be true for more experienced managers, as well, from Joe Maddon to Craig Counsell to Bruce Bochy.

Soon enough, they’ll start filling out the lineup cards for real, now that the start of the exhibition season is about a month away. Batting orders change more than ever in this era, often shaped through the advice (or orders) of front offices, according to the matchups of the day, the strengths and weaknesses of the pitching, hot or cold streaks of those in the order, nagging injuries. Wednesday’s well-considered, well-constructed lineup might be blown up by Thursday afternoon.

Here are some of the more interesting lineup quandaries throughout MLB.

1. Where does Boone place Aaron Judge and Giancarlo Stanton in the Yankees' lineup?

I wrote in the fall about the logic behind Judge hitting leadoff, because of his high on-base percentage and the fact he’d probably get an extra 20 plate appearances or so over the course of the season. But hitting him leadoff also means consigning him to about 20 percent of his plate appearances without a chance of somebody on base.

Judge and Stanton could bat back-to-back, 2 and 3 or 3 and 4, but that would create an opportunity for an opposing manager to deploy right-handers with good sliders -- which has been a challenge for Judge and Stanton in the past. So Boone might be more likely to place a left-handed hitter in between Judge and Stanton -- maybe Greg Bird, perhaps Didi Gregorius -- to extract a toll on any opponent who wants to keep a good right-hander in to face Judge and Stanton.

2. Who will hit second for the Giants?

In a division in which San Francisco faces left-handers Clayton Kershaw, Robbie Ray, Alex Wood, Rich Hill, Patrick Corbin and others, the Giants have sometimes had to battle with a left-handed-heavy lineup. The addition of right-handed hitters Evan Longoria, Andrew McCutchen and Austin Jackson changes that -- and the alterations will be in play on Opening Day, when San Francisco faces Kershaw.

Now Bochy must decide how he will align them. McCutchen will hit near the top of the order, of course, and given Buster Posey's lack of speed, it figures McCutchen could hit third and Posey fourth. Jackson might be the leadoff hitter against lefties.

Bochy figures to want a left-handed hitter to break up that group of right-handed hitters. Because he has always viewed Brandon Crawford as more of a run-producer, Crawford might hit fifth or sixth. That leaves either Joe Panik or Brandon Belt to bat second, and both are good hitters with solid OBP.

It might simply come down to who happens to be swinging well -- if Panik is, he could bat second, with Belt hitting seventh, or vice versa.

3. Where does Mike Trout bat in the Angels’ lineup?

The Angels greatly improved the overall depth of their lineup with the addition of second baseman Ian Kinsler and third baseman Zack Cozart, as well as Justin Upton. Kinsler has hit leadoff for a lot of his career and could fill that role for Anaheim.

After the arrival of Upton last season, Scioscia used Trout in the No. 2 hole, with Upton behind him, presenting a power threat to opposing pitchers. What that’ll mean, of course, is the Angels will start with a string of right-handed hitters, so Scioscia could use the left-handed hitting Kole Calhoun in the cleanup spot, with Cozart, Pujols and maybe lefty Shohei Ohtani in some arrangement.

But Trout could be at No. 2, which more and more seems to be where a lot of teams are placing potent lineup weapons -- as opposed to the prehistoric use of the spot for a guy adept at bunting.

4. Who will bat cleanup for the Red Sox?

Three spots seem set for Boston. Mookie Betts has established himself as the bona fide leadoff man, where his ability to get on base and speed can be put to use. The left-handed hitting Andrew Benintendi will follow the right-handed-hitting Betts. If the Red Sox sign J.D. Martinez -- which is what some rival executives expect, given the limited big-money interest in Martinez in this winter’s market -- then he likely would hit third.

After that, Cora could go with the player swinging the best in the moment, whether it’s Hanley Ramirez, talented third baseman Rafael Devers, or maybe even Dustin Pedroia after he comes back from his knee surgery.

5. Who hits leadoff for the Cubs?

As the Talking Heads said: Same as it ever was. Last year, Maddon used more than half a dozen hitters in the leadoff spot over the course of the season, opening the year with Kyle Schwarber and following with everyone from Jon Jay to Ben Zobrist to -- memorably -- Anthony Rizzo, who batted .300 in 14 games in the No. 1 spot. The Cubs’ leadoff hitters were mediocre, flatly: They tied for 16th in the majors in runs and were 18th in OBP, at .324.

It appears Maddon is set at 2-3-4, with the right-handed-hitting Bryant, the left-handed-hitting Rizzo, and the right-handed-hitting Willson Contreras. The Cubs played their best last season when those three were rolling together in August.

This leaves the same cast of options from which Maddon will choose a leadoff hitter: Schwarber, who has devoted himself to losing weight this offseason and demonstrating that he can be a viable left fielder; Jason Heyward, who had a .326 on-base percentage last year; infielders Javier Baez or Addison Russell; or the center fielder du jour, whether that’s Albert Almora Jr. or Ian Happ -- or maybe Zobrist, if he recovers from a tough 2017.

It seems inevitable Maddon will again consider Schwarber, because of his ability to take a walk. But he’ll have to hit a little more to make that happen, and early on in the season, Maddon might prefer to protect Schwarber from the pressure of the leadoff spot and give him a chance for some sustained early success.

Long winter, long-term ramifications?

There is a thick groundswell of conversation taking place among players about the disastrous free-agent market and how it came to be, with some veterans beginning to discuss the adjustments that need to be made before the players' association prepares for the next round of collective bargaining talks. How that manifests itself in the days and weeks ahead remains to be seen, but some agents and players believe the annual union meetings this spring could be telling.

A few broad lines and suggestions heard on the players’ side:

1. Some agents believe there was not a proper adjustment to the growing number of teams choosing to bypass the winter market and -- fill in the blank for yourself -- tank or rebuild. Jerry Dipoto, the general manager of the Mariners, offered words the other day that perfectly described the current state of this offseason when he said, “You could argue you’re going to compete with more clubs to get the first pick in the draft than you would to win the World Series."

Some folks on management side believe there could be anywhere from eight to 12 teams positioning themselves for the top of the draft in 2019, including the salary-dumping Miami Marlins, and simply declining to spend money on possible improvements. Incredibly, the Marlins have only $137 million in commitments moving forward -- or less than what the Diamondbacks owe Zack Greinke. It is through tanking/rebuilding that tens of millions of dollars that might theoretically be available are not being spent on veteran talent.

In the past, the union has been resistant about a system that forces teams to reach a minimum floor of spending, because the only way that might happen could be through a hard cap.

But agents already are spitballing in earnest about how to deal with the issue of tanking.

2. In the next round of negotiations, it would make sense for union leadership to use a larger group of agents as a sounding board or focus group to draw on their experience about what might work or what might not work. They should provide the boots-on-the-ground perspective for the union, because they talk to teams constantly, they talk to general managers, they have a sense of what’s needed. Within hours after the conclusion of the last CBA talks, some agents who felt shut out of the process precisely forecast the cold free-agent conditions we saw last winter and that we are seeing again this winter -- the lower-than-expected luxury-tax levels serving as a hard cap, the huge volume of free agents working against the veteran players, etc.

3. The players' association might consider constructing a sturdy analytics department, to better match what the union now faces on the team side -- increasingly functional, highly efficient front offices.

Like the Congressional Budget Office, a beefed-up analytics group could serve to better anticipate market trends, such as a movement at a particular position or the expectations for a class of free agents (which is guaranteed to be large again next year because of the enormous number of veterans who will have to settle for one-year contracts before the start of the 2018 season).

A lot of agents do their own statistical work, but individual players and agents alike could draw on a union analytics group that evaluates the market. If a competing agent working to lure a player from a competitor promised a big salary, there would be an independent group the player could pull from to check the reality of the math.

In 1994-95, a common refrain heard around the union-MLB talks was that the union always won because it had the best lawyers. These days, there are longtime agents privately advocating for greater firepower on their side through analysis or negotiating advice.

• Next weekend, J.D. Martinez’s agent, Scott Boras, is scheduled to be in Boston for an event, so it’ll be interesting to see whether Boras and the Red Sox meet about the slugger. Earlier this winter, industry sources indicate, Boston’s first offer to Martinez was $100 million over five years. Arizona also has some degree of interest in bringing back the basher. ... The Indians haven’t spent much money this winter, but they are hopeful about what free-agent signing Melvin Upton Jr. might provide. He’ll probably get an opportunity early in the season, when Michael Brantley and Brandon Guyer will still be working their way back from injury. ... On the podcast the other day, Tim Kurkjian discussed the Hall of Fame voting -- stories about Chipper Jones, Jim Thome, Vladimir Guerrero and Trevor Hoffman -- and the addition of Alex Rodriguez to the Sunday Night Baseball booth; Jesse Rogers talked about the Cubs and the free agency of Jake Arrieta; and a long goodbye to old friend Mark Simon. ... Also: Past podcast conversations with Chipper Jones and Thome.

And today will be better than yesterday.