Adjacent Additions
In a 7-digit numerical code, each group of four consecutive digits adds to 16, and each group of five consecutive digits adds to 19. What is the sum of all 7 digits?
Age
14 to 16
Challenge level
In a 7-digit numerical code, each group of four adjacent digits adds to 16, and each group of five adjacent digits adds to 19.
What is the sum of all seven digits?
Answer: $25$
Using few unknowns
first + (middle 3) = (middle 3) + last $\therefore$ first = last, so the numbers indicated are equal
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The code is:
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Repeat for the 5s:
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The code is:
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Fours: $3a+x=16$
Fives: $4a+x=19$
$\therefore a=3$
All seven: $6a+x = 2a + (4a+x) = 6+19=25$
Using seven unknowns
Let the $7$-digit code be $abcdefg$.
We know that
$(1)$ $a+b+c+d=16$
$(2)$ $b+c+d+e=16$
$(3)$ $c+d+e+f=16$
$(4)$ $d+e+f+g=16$
$(5)$ $a+b+c+d+e=19$
$(6)$ $b+c+d+e+f=19$
$(7)$ $c+d+e+f+g=19$
If we take equation $(1)$ away from equation $(5)$ we obtain $e=3$.
Similarly:
$(6)-(2)$ gives that $f=3$,
$(7)-(3)$ gives that $g=3$,
$(5)-(2)$ gives that $a=3$,
$(6)-(3)$ gives that $b=3$ and
$(7)-(4)$ gives that $c=3$.
Then using equation $(1)$ we find that $d=7$.
So the code is $3337333$ and the sum of the digits is $25$.