Editing Openai/69440b7b-41f8-8010-9331-140a6dd5eed2 (section)

== gpt-5-2-thinking ==
=== User: Think well to give a correct answer, and if possible use two methods to make sure the result is correct. ===
Think well to give a correct answer, and if possible use two methods to make sure the result is correct. Just to make sure you correctly understand the requirements:
* the sides are integer numbers
* the volume is an integer number
* the areas of the sides are irrational
* for (1) find the volume of the tetrahedron with minimum a+b+c+d+e+f
* for (2) find the volume of the tetrahedron that has volume = a+b+c+d+e+f

=== Assistant: I’ll use the standard labeling with vertices A,B,C,DA,B,C,DA,B,C,D and edge-lengths ===
I’ll use the standard labeling with vertices A,B,C,DA,B,C,DA,B,C,D and edge-lengths

AB=a, AC=b, AD=c, BC=d, BD=e, CD=fAB=a,\ AC=b,\ AD=c,\ BC=d,\ BD=e,\ CD=fAB=a, AC=b, AD=c, BC=d, BD=e, CD=f
Face-triangles are: (a,b,d),(a,c,e),(b,c,f),(d,e,f)(a,b,d), (a,c,e), (b,c,f), (d,e,f)(a,b,d),(a,c,e),(b,c,f),(d,e,f).

A face area is rational iff

16 Δ2=(x+y+z)(−x+y+z)(x−y+z)(x+y−z)16\,\Delta^2=(x+y+z)(-x+y+z)(x-y+z)(x+y-z)16Δ2=(x+y+z)(−x+y+z)(x−y+z)(x+y−z)
is a perfect square integer (then Δ=K/4\Delta=\sqrt{K}/4Δ=K/4 is rational). So “irrational face area” ⇔ that KKK is not a square.

Volume check (two independent exact methods):

Method 1 (Gram 3×3):
Let u=AB→,v=AC→,w=AD→u=\overrightarrow{AB}, v=\overrightarrow{AC}, w=\overrightarrow{AD}u=AB,v=AC,w=AD. Then

u ⁣⋅ ⁣v=a2+b2−d22,u ⁣⋅ ⁣w=a2+c2−e22,v ⁣⋅ ⁣w=b2+c2−f22u\!\cdot\! v=\frac{a^2+b^2-d^2}{2},\quad
u\!\cdot\! w=\frac{a^2+c^2-e^2}{2},\quad
v\!\cdot\! w=\frac{b^2+c^2-f^2}{2}u⋅v=2a2+b2−d2,u⋅w=2a2+c2−e2,v⋅w=2b2+c2−f2
and

V2=det⁡(Gram(u,v,w))36.V^2=\frac{\det(\text{Gram}(u,v,w))}{36}.V2=36det(Gram(u,v,w)).
Method 2 (Cayley–Menger 5×5):

288V2=det⁡(0111110a2b2c21a20d2e21b2d20f21c2e2f20).288V^2=\det
\begin{pmatrix}
0&1&1&1&1\\
1&0&a^2&b^2&c^2\\
1&a^2&0&d^2&e^2\\
1&b^2&d^2&0&f^2\\
1&c^2&e^2&f^2&0
\end{pmatrix}.288V2=det0111110a2b2c21a20d2e21b2d20f21c2e2f20.

==== (1) Minimum ====
𝑎
+
𝑏
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𝑐
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𝑑
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𝑒
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𝑓
a+b+c+d+e+f ⇒ report the volume

The minimum total edge length is 28, and one such tetrahedron is:

(a,b,c,d,e,f)=(2,4,5,4,6,7),a+b+c+d+e+f=28.(a,b,c,d,e,f)=(2,4,5,4,6,7),\quad a+b+c+d+e+f=28.(a,b,c,d,e,f)=(2,4,5,4,6,7),a+b+c+d+e+f=28.
Volume (both methods agree):
* Cayley–Menger determinant =10368=288⋅62⇒V=6=10368=288\cdot 6^2\Rightarrow V=6=10368=288⋅62⇒V=6.
* Gram method gives the same V=6V=6V=6.

All four face areas are irrational (each K=16Δ2K=16\Delta^2K=16Δ2 is non-square):
* (a,b,d)=(2,4,4): K=240(a,b,d)=(2,4,4):\ K=240(a,b,d)=(2,4,4): K=240 (not a square)
* (a,c,e)=(2,5,6): K=351(a,c,e)=(2,5,6):\ K=351(a,c,e)=(2,5,6): K=351
* (b,c,f)=(4,5,7): K=1536(b,c,f)=(4,5,7):\ K=1536(b,c,f)=(4,5,7): K=1536
* (d,e,f)=(4,6,7): K=2295(d,e,f)=(4,6,7):\ K=2295(d,e,f)=(4,6,7): K=2295

✅ Answer (1): V=6V=6V=6 (at the minimum possible total edge-length 282828).

==== (2) Find integer-edge tetrahedron with ====
𝑉
=
𝑎
+
𝑏
+
𝑐
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𝑑
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𝑒
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𝑓
V=a+b+c+d+e+f

The smallest solution (by total edge sum, hence by VVV too) I find is:

(a,b,c,d,e,f)=(4,10,17,12,19,22)(a,b,c,d,e,f)=(4,10,17,12,19,22)(a,b,c,d,e,f)=(4,10,17,12,19,22)
a+b+c+d+e+f=84.a+b+c+d+e+f = 84.a+b+c+d+e+f=84.
Volume (both methods agree):
* Cayley–Menger determinant =2032128=288⋅842⇒V=84=2032128=288\cdot 84^2\Rightarrow V=84=2032128=288⋅842⇒V=84.
* Gram method also gives V=84V=84V=84.

All face areas are irrational (non-square KKK’s):
* (4,10,12): K=5616(4,10,12):\ K=5616(4,10,12): K=5616
* (4,17,19): K=15360(4,17,19):\ K=15360(4,17,19): K=15360
* (10,17,22): K=106575(10,17,22):\ K=106575(10,17,22): K=106575
* (12,19,22): K=207495(12,19,22):\ K=207495(12,19,22): K=207495

✅ Answer (2): V=84V=84V=84 (with total edge-length 848484).

If you want, I can also list all (1)-type solutions at sum 282828 (they’re just relabelings) and/or additional equable solutions beyond the minimum.