Is anyone else obsessed with watching R1's thinking output?

Is anyone else obsessed with watching the thinking output?

I find myself watching it and getting anxious when it gets close to the right answer but then does a "oh wait!" and then switches to another direction.

Today, I tried the 1.5B Q5 distil model to see how it did with a simple question:

state density of gold and from this calculate the length of a cube of gold with 1kg mass.

``` Thought for 232 seconds Okay, so I need to figure out the density of gold and then use that to calculate the length of a cube made with 1 kg of gold. Hmm, let's break this down step by step.

First, I remember that density is mass divided by volume. The formula for density is:

Density= Volume Mass ​

But wait, to use this formula, I need to know the volume of the gold cube. Since it's a cube, all sides are equal, right? So if I can find the side length, I can find the volume.

But before that, I need the density of gold. I think gold is really dense, but I'm not exactly sure what that number is. I think it's around 19.3 grams per cubic centimeter. Yeah, that sounds right. Let me confirm that... I think I remember that gold has a density of about 19.3 g/cm³. Okay, so that's the density.

Now, I have a mass of 1 kg. Wait, I need to make sure the units are consistent because density is mass per volume. So, should I convert kilograms to grams? Yeah, that makes sense. 1 kg is 1000 grams.

So, mass is 1000 grams, and density is 19.3 grams per cm³. So, using the density formula, I can rearrange it to solve for volume:

Volume= Density Mass ​

Plugging in the numbers:

Volume= 19.3 grams/cm 3

1000 grams ​

Let me do that division. 1000 divided by 19.3. Hmm, 19.3 goes into 1000 how many times? Let me calculate 19.3 times 50 is 965, which is less than 1000. Then 19.3 times 52 is 1003.6, which is just over 1000. So, it's approximately 52 cm³, maybe around 52.2 cm³ or something.

Wait, actually, let me compute it more accurately. 19.3 multiplied by 52 is 1003.6, which is 3.6 grams over. So, 1000 / 19.3 is approximately 51.88 cm³.

So, the volume is about 51.88 cm³.

Now, since the cube has equal sides, the volume of a cube is side length cubed. So,

Volume=Side 3

Therefore, to find the side length, I need to take the cube root of the volume. So,

Side= 3

Volume ​

Plugging in the volume:

Side= 3

51.88 cm 3

​

Hmm, cube root of 51.88. Let me think about this. 3³ is 27, 4³ is 64, so it's somewhere between 3 and 4 cm. Let me calculate it more accurately.

I know that 3.7³ is 3.7 * 3.7 * 3.7. 3.7 * 3.7 is 13.69, then 13.69 * 3.7. Let me compute that.

13.69 * 3 is 41.07, and 13.69 * 0.7 is 9.583. Adding them together gives 41.07 + 9.583 = 50.653 cm³. Hmm, that's close to 51.88, but a bit less.

So, 3.7 cm³ gives 50.653 cm³. The volume we have is 51.88 cm³, so it's a bit higher. So, let's try 3.72 cm.

3.72 * 3.72 = ?

First, 3.7 * 3.7 = 13.69, as before. Then, 0.02 * 3.72 is 0.0744, so 3.72 * 3.72 = 13.69 + 0.0744 + 0.0744? Wait, no, that's not correct. Actually, (a + b)² = a² + 2ab + b². So, (3.7 + 0.02)² = 3.7² + 23.7 0.02 + 0.02² = 13.69 + 0.148 + 0.0004 = 13.8384.

Then, 13.8384 * 3.72. Let me compute that.

13.8384 * 3 = 41.5152

13.8384 * 0.7 = 9.68688

13.8384 * 0.02 = 0.276768

Adding them together:

41.5152 + 9.68688 = 51.20208

51.20208 + 0.276768 = 51.478848 cm³

So, 3.72³ is approximately 51.478848 cm³, which is a bit less than 51.88 cm³.

So, let's try 3.73.

3.73 * 3.73 = ?

Again, using the same method:

3.73 is 3.7 + 0.03

So, (3.7 + 0.03)² = 3.7² + 23.7 0.03 + 0.03² = 13.69 + 0.222 + 0.0009 = 13.9129

Then, 13.9129 * 3.73.

Compute step by step:

13.9129 * 3 = 41.7387

13.9129 * 0.7 = 9.73903

13.9129 * 0.03 = 0.417387

Adding them up:

41.7387 + 9.73903 = 51.47773

51.47773 + 0.417387 = 51.895117 cm³

Wow, that's very close to 51.88 cm³. So, 3.73³ is approximately 51.895117 cm³.

Which is a little over 51.88 cm³. So, the exact cube root is just a little less than 3.73 cm. Maybe 3.728 cm?

Wait, perhaps I should use linear approximation or Newton-Raphson method for a better estimate. But this is getting complicated. Maybe I can use a calculator here, but since I don't have one, perhaps I can estimate.

Alternatively, maybe I can use the exact calculation.

But actually, perhaps I can note that 3.72³ is 51.478848, and 3.73³ is 51.895117. Since 51.88 is between them, so the cube root is somewhere in between.

The difference between 51.88 and 51.478848 is 0.401152.

The difference between 51.895117 and 51.478848 is 0.416269.

So, the fraction is 0.401152 / 0.416269 ≈ 0.962.

So, the cube root is approximately 3.72 + 0.962*(0.001) ≈ 3.72 + 0.000962 ≈ 3.720962 cm.

So, approximately 3.721 cm.

Therefore, the side length is approximately 3.721 cm.

But wait, let me double-check my calculations because this seems a bit tedious.

Alternatively, maybe I can use the formula for the volume of a cube and solve for the side length.

Wait, another approach: let's take the cube root of 51.88.

We know that 3.7³ is 50.653, and 3.72³ is 51.4788, which is less than 51.88.

So, 3.72³ = 51.4788

Let me compute 3.72 + delta, where delta is small, such that (3.72 + delta)³ = 51.88

Using the binomial expansion:

(3.72 + delta)<sup>3</sup> = 3.72³ + 3(3.72)²delta + 3 (3.72)delta² + delta³

We can ignore delta squared and delta cubed since delta is small.

So,

(3.72 + delta)<sup>3</sup> ≈ 3.72³ + 3*(3.72)²delta = 51.4788 + 3 (13.8384)delta = 51.4788 + 41.5152 delta

Set this equal to 51.88:

51.4788 + 41.5152*delta = 51.88

Subtract 51.4788:

41.5152*delta = 0.4012

So,

delta ≈ 0.4012 / 41.5152 ≈ 0.00962

So, delta ≈ 0.00962 cm.

Therefore, the cube root is approximately 3.72 + 0.00962 ≈ 3.72962 cm.

So, approximately 3.73 cm.

But wait, when I cube 3.73, I get 51.895 cm³, which is just a bit over 51.88. So, actually, the cube root is slightly less than 3.73. Let me denote delta as negative.

So,

(3.72 + delta)<sup>3</sup> = 51.88

We found that 3.72³ is 51.4788, and 3.73³ is 51.8951.

So, 51.88 is 51.8951 - 0.0151 above 3.73³.

So, delta is such that:

51.4788 + 41.5152*delta = 51.88

So, 41.5152*delta = 51.88 - 51.4788 = 0.4012

Therefore,

delta = 0.4012 / 41.5152 ≈ 0.00962

But since 3.73³ is higher, delta is negative, so delta ≈ -0.00962 cm.

So, the cube root is approximately 3.72 - 0.00962 ≈ 3.71038 cm.

Therefore, approximately 3.7104 cm.

So, the side length is approximately 3.71 cm.

Wait, let me verify by cubing 3.71:

3.71 * 3.71 = 13.7641

13.7641 * 3.71:

Let me compute 13.7641 * 3 = 41.2923

13.7641 * 0.7 = 9.63487

13.7641 * 0.01 = 0.137641

Adding them together:

41.2923 + 9.63487 = 50.92717

50.92717 + 0.137641 ≈ 51.06481 cm³

Hmm, 51.06481, which is still less than 51.88. So, 3.71³ is 51.06481 cm³.

So, that's still less than 51.88. Let me try 3.72:

As before, 3.72³ is 51.4788 cm³, which is still less.

3.721³: Let's compute 3.721³.

First, 3.721 * 3.721:

3.721 * 3.721

Compute 3 * 3.721 = 11.163

0.7 * 3.721 = 2.6047

0.02 * 3.721 = 0.07442

0.001 * 3.721 = 0.003721

So, adding them:

11.163 + 2.6047 = 13.7677

13.7677 + 0.07442 = 13.84212

13.84212 + 0.003721 ≈ 13.845841

So, 3.721² ≈ 13.845841

Now, 13.845841 * 3.721:

Again, break it down:

13 * 3.721 = 48.373

0.845841 * 3.721 ≈ Let's compute 0.8 * 3.721 = 2.9768

0.04 * 3.721 = 0.14884

0.005841 * 3.721 ≈ 0.02176

Adding them together:

2.9768 + 0.14884 = 3.12564

3.12564 + 0.02176 ≈ 3.1474

So, total is 48.373 + 3.1474 ≈ 51.5204 cm³

So, 3.721³ ≈ 51.5204 cm³

Which is still less than 51.88. Hmm, so 3.721³ is 51.5204, which is less than 51.88. So, let's try 3.722.

Compute 3.722³:

First, 3.722²:

Again, 3.72² is 13.8384, so 3.722² = 13.8384 + 2(0.002)3.72 + (0.002)<sup>2</sup>

Wait, actually, 3.722 is 3.72 + 0.002, so:

(3.72 + 0.002)² = 3.72² + 23.72 0.002 + 0.002² = 13.8384 + 0.01488 + 0.000004 = 13.853284

Then, 13.853284 * 3.722:

Break it down:

13 * 3.722 = 48.386

0.853284 * 3.722 ≈ Let's compute:

0.8 * 3.722 = 2.9776

0.05 * 3.722 = 0.1861

0.003284 * 3.722 ≈ 0.01222

Adding them:

2.9776 + 0.1861 = 3.1637

3.1637 + 0.01222 ≈ 3.1759

Total volume: 48.386 + 3.1759 ≈ 51.5619 cm³

So, 3.722³ ≈ 51.5619 cm³, which is less than 51.88.

So, 3.722³ = 51.5619

Difference from 51.88: 51.88 - 51.5619 = 0.3181

So, need to add more.

Let me compute the derivative of the volume with respect to side length to approximate the delta.

V = s³

dV/ds = 3s²

At s = 3.722, dV/ds = 3(3.722)² ≈ 313.853284 ≈ 41.55985

So, delta V ≈ dV/ds * delta = 41.55985 * delta = 0.3181

So, delta ≈ 0.3181 / 41.55985 ≈ 0.00765 cm

Therefore, s ≈ 3.722 + 0.00765 ≈ 3.72965 cm

So, approximately 3.7297 cm

So, the cube root is approximately 3.7297 cm

Therefore, the side length of the cube is approximately 3.73 cm

To be precise, about 3.73 cm.

Therefore, the edge length of the cube is approximately 3.73 cm.

Let me verify by cubing 3.73:

3.73 * 3.73 = 13.9129

13.9129 * 3.73:

13 * 3.73 = 48.49

0.9129 * 3.73 ≈ Let's compute:

0.9 * 3.73 = 3.357

0.0129 * 3.73 ≈ 0.0482

Adding together: 3.357 + 0.0482 ≈ 3.4052

Total volume: 48.49 + 3.4052 ≈ 51.8952 cm³

Which is very close to 51.88 cm³.

So, 3.73 cm³ is approximately 51.8952, which is about 51.90, which is very close to 51.88.

Therefore, 3.73 cm is a good approximation.

So, the edge length is approximately 3.73 cm.

Hence, the edge of the cube is approximately 3.73 cm.

Final Answer The edge of the cube is approximately \boxed{3.73} cm. ```