data/problems/315.yml
---
:id: 315
:name: Digital root clocks
:url: https://projecteuler.net/problem=315
:content: " \n\nSam and Max are
asked to transform two digital clocks into two \"digital root\" clocks. \nA digital
root clock is a digital clock that calculates digital roots step by step.\n\nWhen
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. \nFor 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.\n\nEvery digital
number consists of some light segments: three horizontal (top, middle, bottom) and
four vertical (top-left, top-right, bottom-left, bottom-right). \nNumber \" **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.\n\nThe clocks consume energy only when segments are turned on/off.
\ \nTo turn on a \" **2**\" will cost 5 transitions, while a \" **7**\" will cost
only 4 transitions.\n\nSam and Max built two different clocks.\n\nSam'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.
\ \nFor the example, with number 137, Sam's clock requires:\n\n| \" **137**\" |
: | (2 + 5 + 4) × 2 = 22 transitions (\" **137**\" on/off). |\n| \" **11**\" | :
| (2 + 2) × 2 = 8 transitions (\" **11**\" on/off). |\n| \" **2**\" | : | (5) ×
2 = 10 transitions (\" **2**\" on/off). |\n\nFor a grand total of 40 transitions.\n\nMax'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. \nFor
number 137, Max's clock requires:\n\n| \" **137**\" \n \n | : \n \n | 2 + 5
+ 4 = 11 transitions (\" **137**\" on) \n7 transitions (to turn off the segments
that are not needed for number \" **11**\"). |\n| \" **11**\" \n \n \n | : \n
\ \n \n | 0 transitions (number \" **11**\" is already turned on correctly) \n3
transitions (to turn off the first \" **1**\" and the bottom part of the second
\" **1**\"; \nthe top part is common with number \" **2**\"). |\n| \" **2**\"
\ \n \n | : \n \n | 4 transitions (to turn on the remaining segments in order
to get a \" **2**\") \n5 transitions (to turn off number \" **2**\"). |\n\nFor
a grand total of 30 transitions.\n\nOf course, Max's clock consumes less power than
Sam's one. \nThe two clocks are fed all the prime numbers between A = 10<sup>7</sup>
and B = 2×10<sup>7</sup>. \nFind the difference between the total number of transitions
needed by Sam's clock and that needed by Max's one.\n\n"