How much salt catches the cube?

In one line: a little salt near the string can work, and a big heap right on the string stops it freezing. The exact grams don't matter. In the first minute the top of a cube can only melt about 1.1 g of ice, and that water can only dissolve so much salt: somewhere between about 0.28 g and 0.76 g on this cube's top (between 1/32 and 1/16 tsp, up to 1/8 tsp), maybe less. Salt that dissolves in time drains off and leaves chilled ice that freezes new ice round the string. Salt that's still next to the string at the pull keeps the water there too salty to freeze.

Salt makes the water freeze colder

Pure water freezes at 0 °C. Water with salt dissolved in it has to get colder before it freezes: the saltier, the colder. The graph is the measured curve (Bodnar 1993; ASHRAE's brine table agrees to within about half a degree). It stops at 23.2% salt and −21.2 °C. That's the coldest salt water can get and still stay liquid next to ice; saltier than that and the extra salt just sits there undissolved.

So when salt lands on the wet top of a cube that's at 0 °C, it dissolves into the thin film of water, and now that water "wants" to be much colder than 0 °C before it freezes. The ice it touches is too warm to stay ice next to it, so the ice melts, and the new meltwater dissolves more salt. The model holds the film at the saltiest and coldest it can be, 23.2% salt at −21.2 °C, all the time there's salt left. That's the most it could be: really, between the grains, fresh meltwater waters the film down faster than salt can spread into it, so it's a bit warmer and melts less. So the melting, the stored cold and the new ice in the sim are the most they could be.

Melting takes heat, and the heat comes from the ice

• Melting 1 g of ice takes 333 joules (IAPWS, the international standard for water and ice). That heat has to come from somewhere, and in a minute the only places close enough are the ice itself and the bowl water around the cube.
• The new salty water also has to be cooled down to −21.2 °C, and the salt takes a little heat to dissolve. Put together, each gram of ice that melts pulls about 262 J out of the ice underneath (333 to melt it, minus 92 J because the brine it makes is that cold, plus 20 J to dissolve its salt).
• Taking heat out of ice makes it colder, starting at the top. Ice passes heat along slowly (conductivity 2.14 W per metre per degree, Slack 1980). Through the top face alone, a minute's worth of heat is the same as cooling a layer about 9.2 mm thick by the full 21 degrees. Really the chill fades off gradually with depth: the drawing's lines show where the ice is colder than −1 °C and −10 °C, and after a minute of salt the −1 °C line is about 16 mm down in the middle.
• The cube floats in ice water at 0 °C, so its sides and bottom touch water at 0 °C. As the ice gets colder than that, the bowl water freezes onto the outside and gives its latent heat to the cube. In the model that's where most of the heat comes from after a while: about 66% of the 340 J pulled out in the first minute. That share depends on the wait and on the strip of side above the water: it's about 51% if you stop at 30 s and 76% at 2 minutes, with that strip taken as dry. It gets there by a short path: the cold top is only about 2 mm from the water at the rim. The skin of bowl ice is drawn as a thin orange-edged layer on the sides, as if it were even (really it's thicker near the top).

So there's a limit, but only roughly

With the salt never running out, the top of the cube melts about 1.1 g of ice in the minute (the top face goes down evenly by about 2.5 mm). Salty water at its saltiest holds 0.30 g of salt per gram of water, so that meltwater can dissolve about 0.34 g of salt, or about 0.36 g counting the salt's own weight (it lowers the cube a little, see below). That's the model's middle answer, and it's softer than it looks:

• The rim. The model's middle case treats the strip of side sticking out of the water (1.85 mm, or less with salt on top) as dry. But water climbs a wet wall by a few millimetres, so it may be wet right up to the top edge. Then heat gets to the cold top faster, and the minute dissolves about 0.58 g, and more when the model's blocks are made smaller (about 0.76 g with blocks about a quarter to a third of the size, and still creeping up).
• The film is the coldest it could be (see above), so the real melting is slower.
• Brine that doesn't drain. The model lets the salty water run off over the edge into the bowl at once. Really some stays in the string, in any pit in the ice and between the grains (a heap of poured salt is almost half air), and keeps the film salty for longer.

The last two push the answer down by amounts the model can't size. For the bottom of the range the model uses a rim drier than it can really be, which gives about 0.28 g; the real answer could be lower still. So the sim doesn't give a sharp line. It says:

• Likely dissolves in time: new ice can form round the string. It all dissolves in time even at the bottom of the range. Whether that new ice then holds the cube, the sim can't say.
• Could go either way: the amount is inside the range, so it depends on things the model can't pin down.
• Likely too much, the film stays too salty: more than even the top of the range. With the salt spread over the top, the slider doesn't get there: a floating cube can't carry that much (see below). A heap on the string can.
• Dissolves, but only a thin hug: it dissolves so fast that the new ice is under half as thick as the most the model gives. That "half" line is our choice, not physics: the sim has to call a hug "thin" somewhere, and nothing in the model says where grip starts.

The wait matters a lot: the amount the minute can dissolve grows almost in step with the wait, from about 0.19 g at 30 s to 0.34 g at a minute and 0.65 g at 2 minutes (middle case, no salt weight). Twice the wait, nearly twice the salt, because most of the heat comes in from the bowl across that short path at the rim, and that flow soon settles down to a steady rate.

The salt's weight

The cube floats with only about 1.85 mm of it sticking out. Salt on top weighs it down: each gram pushes it about 2.0 mm lower. It can carry only about 0.91 g before its top goes under water (that's the difference between how heavy water and ice are, times the cube's size). That's why the slider stops at 1/8 tsp (0.75 g). A bigger pile wouldn't stay on a floating cube this size: the top would go under and the bowl water would wash the salt off. The model counts the weight: a lower cube has less dry side, so heat reaches the top a bit faster.

Where the salt lands

The slider is the salt that stays on this cube's top, near the string, and the model spreads it evenly over the whole 2.2 cm top. Salt that misses the cube, or slides off into the bowl, doesn't count: a whole bowl of water only gets a little salty. That's how the note "The kid can dump salt everywhere — that was fine" can fit: probably most of a dump ends up in the bowl, and what matters is how much sits right under the string. (We haven't asked yet whether those cubes still got caught.)

A heap right on the string is worse than the same salt spread out. The minute can only dissolve about 0.057 to 0.15 g for each square centimetre of salted top, on average over the top (the model spreads the melting evenly, and much of the heat comes in near the rim, so it can't say how much reaches the middle, where the string lies), so a heap has much more salt over each bit of ice (a small patch does get a little extra heat from the ice around it). A big heap on the string means salt is still next to the string at the pull, and nothing freezes there. That's the page's "mountain of salt" and "too much salt = puddle". The model can't do heaps, so it can't say exactly how big a heap is too big.

The hug

Once the salt has all dissolved, fresh water wets the top again. The model assumes the salty water drains off over the edge into the bowl at once; really some stays in the string and in any pit, which delays this. That water is at 0 °C, but the ice under it is now much colder, so the water freezes onto the top, and round the bottom of the string. That new ice is what the string is caught in.

• The most on the slider (1/32 tsp) gives up to 0.11 g of new ice: a layer up to 0.25 mm thick, 13% of the way up the 2 mm string. The rest of the stored cold warms back up from the bowl side.
• Too little salt (1/512 tsp) is all gone after less than a second. The ice only had that long to chill, so it stores about 10 J of cold (a full minute stores about 73 J) and freezes up to about 0.027 g of new ice, about 0.059 mm thick.
• Salt that dissolves just before the pull leaves little time for the new ice to form, so the hug is thinner there too.

What the sim can't tell you

• Whether the string holds. The model gives how much new ice forms, not how strongly it grips wet twine. When the string has frozen in and you pull, the sim shows the ice holding: the cube comes up with the string, and the pull it takes (nothing while it floats, rising to its whole weight, about 10 g, once it is out of the water) is weight minus buoyancy. Whether real ice holds, the model can't say. A 0.25 mm hug round a 2 mm string might hold a 10 g cube or snap; the notes say about 1 in 4 tries worked.
• Exact grams and timing. See the range above. The seconds are the model's, with the salt dissolving instantly, spread evenly over the whole top, and the salty water running off at once. Treat them as rough.
• Heaps. The model only does salt spread evenly over the top.
• Cubes straight from the freezer. The model's cube floats in ice water at 0 °C. A cube at −18 °C from the freezer would already hold a lot of cold, and would freeze the string on with no salt at all.

Assumptions: a 7/8 inch restaurant ice-machine cube floating in ice water at 0 °C. The real Manitowoc "Dice" cube is a slightly slanted block that weighs about 9.45 g (48 to the pound); the model uses a true cube 2.22 cm on a side, 10.1 g. Level US teaspoons of table salt at 6.0 g each (USDA); the page's wait of about a minute; 2 mm cotton twine; air on top ignored (it would add about 1 to 12% more heat).