On the Subject of Hyperage

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Module contains of 36 buttons and hypercube.

• To solve it, find hypervolume of hyperparallelepiped Γ.
• Γ is defined by one of its points: O(0,0,0,0).
• Γ is defined by four of its sides: OA(x1,y1,z1,w1); OB(x2,y2,z2,w2); OC(x3,y3,z3,w3); OD(x4,y4,z4,w4).

Finding correct squares

Hovering the cursor over one of the buttons will:

1. Show a rotation of hypercube.
   • Rotation is notated as two letters. Rotation AB means that after rotation, axis +A transformed to axis +B.
   • Notice: if you're looking directly at buttons, then:
     • +X is left to right,
     • +Y is back to front,
     • +Z is bottom to top,
     • +W is inner to outer.
2. Show 4 values, one of them will be unknown and showed like this: ?.???.
3. Show base-36 digit that is assigned to the button. It is shown just for convinience.

Find 4 buttons that form a square (that will be called square buttons), center of which is exactly on the center of any other button (that will be called center button) but also their rotations are IJ, JK, KL, LI in clockwise order, where I, J, K, L are non-repeated axis of hypercube (X, Y, Z, W) in any order.

Find first square button in reading order. Its coordinates is A. Coordinates of the next button clockwise is B and so on. Center buttons coordinates are coordinates of A, B, C and D that are missing. Replace the only unknown coordinate with 1.000.

To calculate the hypervolume of Γ, compute the absolute value of determinant of matrix, rows of which contain the coordinates of A, B, C and D in XYZW-order.

Convert the hypervolume to base-6 and truncate the result to 3 digits after point.

Submission

After pressing any button, the module will go in submission state. Initially, the number on the center will be 0.

• Pressing any numbered button will add the digit to the answer string (except when the center number is 0, then digit will replace the center number).
• Pressing any button of the word "RESET" will clear the input.
• Pressing button with "." on it will add the dot to the number.
• Pressing any button of the word "SUBMIT" will submit current input. Notice: module will always expect three digits after dot, no more, no less.
  • If the answer was correct, module will shortly solve.
  • If the answer was wrong, module will return to initial state and issue a strike. All the coordinates and rotations will be the same.
• Pressing non-labeled button will do nothing.

Be aware that after every press, buttons may change their positions in predictable fashion.