Back to the future Doc. Brown's Accelerating Reel puzzle

[Tito_H]

When scrambling this puzzle, I wondered whether any sequence of numbers could be solved.

In the formula for Step 1, all numbers except for J=0 are divided by 2, so the dividend must be even. In formulas B through I, the dividend is the sum of every other number in the range 'a' through 'j'.

The sum of two numbers can be calculated in four ways, with ① and ② resulting in an even sum.

① Even + Even = Even number
② Odd + Odd = Even number
③ Even + Odd = Odd number
④ Odd + Even = Odd number

From the above, there are following 4 possible scramble sequences that can be solved:
(X for even numbers and Y for odd numbers.)

a b c d e f g h i j
1. X X X X X X X X X X
2. Y Y Y Y Y Y Y Y Y Y
3. X Y X Y X Y X Y X Y
4. Y X Y X Y X Y X Y X

For example, the scramble for the following sequence of numbers does not fit any of the four patterns listed above, and therefore cannot be solved.

a b c d e f g h i j
0 0 0 0 0 0 1 0 0 0 → not solvable
1 1 1 1 2 1 1 1 1 1 → not solvable
0 1 0 1 0 1 0 1 0 0 → not solvable
1 0 1 0 1 0 1 0 1 1 → not solvable

Please be careful if you are creating your own Scramble sequences.

 

[Tito_H]

I finally managed to solve one of the two problems I set out to tackle (below).

- Solution for base-9 reels numbered 0-8 instead of 0-9.

Luckily, this solution didn't have to be too different from the previous one.

<Premise>
This time the puzzle has a total of 9 reels, with numbers 0 to 8 on each reel.

<Definition>
The 9 buttons are represented by the following letters and each contains the number of times it has been pressed.

A B C D E F G H I

The 9 numbers are represented by the following alphabets.

a b c d e f g h i

<Calculation>
Step 1. Calculate the number of times you need to press the buttons A to I using the formula below.

A = - ((i - h) + (b - a)) / 2
B = b - (a + c) / 2
C = c - (b + d) / 2
D = d - (c + e) / 2
E = e - (d + f) / 2
F = f - (e + g) / 2
G = g - (f + h) / 2
H = h - (g + i) / 2
I = 0

Step 2. If there are numbers of times obtained in step 1 above with between -8 and -5 or 5 and 8, add or subtract 9 respectively.

-8 → 1 (↓ 9 is added)
-7.5 → 1.5
-7 → 2
-6.5 → 2.5
-6 → 3
-5.5 → 3.5
-5 → 4
5 → -4 (↓ 9 is subtracted)
5.5 → -3.5
6 → -3
6.5 → -2.5
7 → -2
7.5 → -1.5
8 → -1

Step 3. After step 2 above, if there are columns of decimals, add 4.5 if the numbers are negative and subtract 4.5 if the numbers are positive.

In addition, in the formula in Step 1, 'a' through 'i' each appear twice in separate formulas, so it is guaranteed that the number of times the calculation results in an odd number will always appear evenly.

-4.5 → 0 (↓ 4.5 is added)
-3.5 → 1
-2.5 → 2
-1.5 → 3
-0.5 → 4
0.5 → -4 (↓ 4.5 is subtracted)
1.5 → -3
2.5 → -2
3.5 → -1
4.5 → 0

That's it!
There is no Step4 this time.

Then, let's solve it with an example.

a b c d e f g h i
7 0 2 4 2 4 2 5 3 → Scrambled

Step 1:
A = ((h - i) + (a - b)) / 2 = ((5 - 3) + (7 - 0)) / 2 = 4.5
B = b - (a + c) / 2 = 0 - (7 + 2) / 2 = -4.5
C = c - (b + d) / 2 = 2 - (0 + 4) / 2 = 0
D = d - (c + e) / 2 = 4 - (2 + 2) / 2 = 2
E = e - (d + f) / 2 = 2 - (4 + 4) / 2 = -2
F = f - (e + g) / 2 = 4 - (2 + 2) / 2 = 2
G = g - (f + h) / 2 = 2 - (4 + 5) / 2 = -2.5
H = h - (g + i) / 2 = 5 - (2 + 3) / 2 = 2.5
I = 0

Step 2:
Not aplicable

Step 3:
There are 4 columns (A, B, G, and H) applicable.

A: 4.5 → 0
B: -4.5 → 0
C: 0
D: +2
E: -2
F: +2
G: -2.5 → +2
H: +2.5 → -2
I: 0

<Conclusion>
The following buttons should be pushed to solve this case:

D button 2 times
E button + Yellow button 2 times
F button 2 times
G button 2 times
H button + Yellow button 2 times

Now, press each button to see if the numbers line up.

a b c d e f g h i
7 0 2 4 2 4 2 5 3 → Scrambled
↓ D +2
4 4 4 4 4 8 8 4 4
↓ E -2
5 7 0 2 4 6 4 7 5
↓ F +2
6 6 6 6 6 6 6 2 2
↓ G +2
0 7 5 3 1 8 6 4 6
↓ H -2
4 4 4 4 4 4 4 4 4 → Solved

Surely they've lined up.

Let's try one more.

(Scramble)
7 7 7 7 7 7 7 7 7
↓ F -4 times
5 0 4 8 3 7 3 8 4
↓ C +3 times
2 3 4 2 0 7 6 5 4
↓ I +5 times
6 2 7 0 2 4 7 1 4 → Scrambled

Step1:
A = ((h - i) + (a - b)) / 2 = ((1 - 4) + (6 - 2)) / 2 = 0.5
B = b - (a + c) / 2 = 2 - (6 + 7) / 2 = -4.5
C = c - (b + d) / 2 = 7 - (2 + 0) / 2 = 6
D = d - (c + e) / 2 = 0 - (7 + 2) / 2 = -4.5
E = e - (d + f) / 2 = 2 - (0 + 4) / 2 = 0
F = f - (e + g) / 2 = 4 - (2 + 7) / 2 = -0.5
G = g - (f + h) / 2 = 7 - (4 + 1) / 2 = 4.5
H = h - (g + i) / 2 = 1 - (7 + 4) / 2 = -4.5
I = 0

Step 2:
There is one column applicable.

A: +0.5
B: -4.5
C: +6 → -3
D: -4.5
E: 0
F: -0.5
G: +4.5
H: -4.5
I: 0

Step 3:
There are six columns applicable.

A: 0.5 → -4
B: -4.5 → 0
C: -3
D: -4.5 → 0
E: 0
F: -0.5 → +4
G: 4.5 → 0
H: -4.5 → 0
I: 0

<Conclusion>
The following buttons should be pushed to solve this case:

A button + Yellow button 4 times
C button + Yellow button 3 times
F button 4 times

Now, press each button to see if the numbers line up.

a b c d e f g h i
6 2 7 0 2 4 7 1 4 → Scrambled
↓ A -4
6 7 8 6 4 2 1 0 8
↓ C -3
0 4 8 3 7 2 7 3 8
↓ F +4
2 2 2 2 2 2 2 2 2 → Solved

Surely they've lined up.

This concludes the solution to this puzzle in base 9 (0-8).

This Doc's puzzle was well worth the effort. Thank you, Doc.

 

[Tito_H]

I looked up the God number for this base-9 reels. The numbers whose absolute value does not decrease even after Steps 1 to 3 are +4 or -4. Therefore, we can infer that there is a sequence that cannot be solved unless you press the buttons as shown below, and that the God number is 32.

A: +4 or -4
B: +4 or -4
C: +4 or -4
D: +4 or -4
E: +4 or -4
F: +4 or -4
G: +4 or -4
H: +4 or -4
I: 0

Total: 32 times

Let's actually create a sequence that can be solved with God number.

(Scramble)
A B C D E F G H I
7 7 7 7 7 7 7 7 7
↓ A: -4
7 3 8 4 0 5 1 6 2
↓ B: -4
3 3 4 5 6 7 8 0 1
↓ C: -4
4 8 4 1 7 4 1 7 4
↓ D: -4
1 0 0 1 3 5 7 0 2
↓ E: -4
3 6 1 6 3 1 8 6 4
↓ F: -4
1 8 7 7 8 1 4 7 1
↓ G: -4
4 6 0 4 0 6 4 3 2
↓ H: -4
3 0 7 6 6 7 0 3 7 → Scrambled

This sequence (3 0 7 6 6 7 0 3 7) is an example of a number that can be solved with God number.

Let's solve it with my solution.

(Calculation)
A B C D E F G H I
3 0 7 6 6 7 0 3 7 → Scrambled

Step1:
Calculate A to I.

A = ((h - i) + (a - b)) / 2 = ((3 - 7) + (3 - 0)) / 2 = -0.5
B = b - (a + c) / 2 = 0 - (3 + 7) / 2 = -5
C = c - (b + d) / 2 = 7 - (0 + 6) / 2 = 4
D = d - (c + e) / 2 = 6 - (7 + 6) / 2 = -0.5
E = e - (d + f) / 2 = 6 - (6 + 7) / 2 = -0.5
F = f - (e + g) / 2 = 7 - (6 + 0) / 2 = 4
G = g - (f + h) / 2 = 0 - (7 + 3) / 2 = -5
H = h - (g + i) / 2 = 3 - (0 + 7) / 2 = -0.5
I = 0

Step 2:
There are 2 columns applicable.

A: -0.5
B: -5 → +4
C: +4
D: -0.5
E: -0.5
F: +4
G: -5 → +4
H: -0.5
I: 0

Step 3:
There are 4 columns applicable.

A: -0.5 → +4
B: +4
C: +4
D: -0.5 → +4
E: -0.5 → +4
F: +4
G: +4
H: -0.5 → +4
I: 0

<Conclusion>
The following buttons should be pushed to solve this case:

From A to H buttons 4 times each (= 32 times)

This is exactly the same as the estimated God number (32 times) presented at the beginning.

What surprised me was that the God number went from 19 on the base-10 reels to a massive 32 on the base-9 reels. I find it hard to believe that we really have to press the blue buttons 32 times to solve this sequence (3 0 7 6 6 7 0 3 7), although that's what happens with my solution...

 

[Tito_H]

I have completed this puzzle by Scratch programming method.

https://scratch.mit.edu/projects/1314983662