Digital Root Clocks

Sam and Max are asked to transform two digital clocks into two “digital root” clocks.
A digital root clock is a digital clock that calculates digital roots step by step.
When a clock is fed a number, it will show it and then it will start the calculation, showing all the intermediate values until it gets to the result.
For example, if the clock is fed the number 137, it will show: “137” → “11” → “2” and then it will go black, waiting for the next number.
Every digital number consists of some light segments: three horizontal (top, middle, bottom) and four vertical (top-left, top-right, bottom-left, bottom-right).
Number “1” is made of vertical top-right and bottom-right, number “4” is made by middle horizontal and vertical top-left, top-right and bottom-right. Number “8” lights them all.
The clocks consume energy only when segments are turned on/off.
To turn on a “2” will cost 5 transitions, while a “7” will cost only 4 transitions.
Sam and Max built two different clocks.
Sam’s clock is fed e.g. number 137: the clock shows “137”, then the panel is turned off, then the next number (“11”) is turned on, then the panel is turned off again and finally the last number (“2″) is turned on and, after some time, off.
For the example, with number 137, Sam’s clock requires:”137”
(2 + 5 + 4) × 2 = 22 transitions (“137” on/off).
(2 + 2) × 2 = 8 transitions (“11” on/off).
(5) × 2 = 10 transitions (“2″ on/off).

For a grand total of 40 transitions.

Max’s clock works differently. Instead of turning off the whole panel, it is smart enough to turn off only those segments that won’t be needed for the next number.
For number 137, Max’s clock requires:”137”
2 + 5 + 4 = 11 transitions (“137” on)
7 transitions (to turn off the segments that are not needed for number “11”).
0 transitions (number “11” is already turned on correctly)
3 transitions (to turn off the first “1” and the bottom part of the second “1”;
the top part is common with number “2”).
4 transitions (to turn on the remaining segments in order to get a “2”)
5 transitions (to turn off number “2”).

For a grand total of 30 transitions.

Of course, Max’s clock consumes less power than Sam’s one.
The two clocks are fed all the prime numbers between A = 107 and B = 2×107.
Find the difference between the total number of transitions needed by Sam’s clock and that needed by Max’s one.

To provide you with a high-level approach to solve this.

1. First, the prime numbers between A and B need to be generated.
Methods for generating prime numbers are abundant for coding, including the Sieve of Eratosthenes algorithm.

2. Then, you need to set the cost of each digit representation on the digital clock.

digit_cost = {0:6, 1:2, 2:5, 3:5, 4:4, 5:5, 6:6, 7:4, 8:7, 9:6}

3. Define a function for the digital root.

In each iteration, you’re evaluating the difference in cost between Sam’s clock and Max’s clock. For Sam, the cost includes turning on and off each digit in the number (i.e., the digital root calculation) and for Max, the cost includes only changing the segments from one digit to the next.

Sam’s cost is simply the sum of the costs of each digit in each step of the transformation.

Max’s cost, which requires a transition model, is a bit harder to compute. One way to calculate the transition cost is to represent each digit as a binary number (where a binary digit represents a segment on the clock, for example, 8 would have all segments). For each step, you’d calculate how many segments (binary digits) have changed from the previous digit and add up these changes for the total cost.

4. Iterate over the list of prime numbers, compute the transition cost of Sam’s and Max’s clocks for each prime number, and keep track of the total cost for both clocks.

5. Finally, calculate the difference between Sam’s and Max’s total costs.

Using these methods would more likely require a programming language like Python rather than conventional math.

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