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Star Sum

Stage: 3 Short Challenge Level: Challenge Level:2 Challenge Level:2

The ten numbers in the star total 75.

Each number appears in two "lines".

And there are five lines, making a total of 150. Which implies the total for each line is 30.

This means that R+I=24 and therefore R=11 and I = 13 (or vice versa).

If R=11 then N=12, H =10 and C=11 which is impossible.

If R=13 then N = 10; H = 12 and C= 9, which is correct.

Alternatively:
once you know that each line adds up to 30 we can set up the following equations:
$(1)    4+N+R+3=30 ⇒N+R=23$
$(2)    1+R+I+5=30 ⇒R+I=24$
$(3)    3+I+C+7=30 ⇒I+C=20$
$(4)    4+H+C+5=30 ⇒H+C=21$
$(5)    1+N+H+7=30 ⇒N+H=22$
Then we are going to use a trick to “isolate R” by alternately adding and subtracting equations to get what we want, so consider
$(N+R)-(N+H)+(H+C)-(I+C)+(R+I)=23-22+21-20+24$
$⇒2R=26 ⇒R=13.$
 

This problem is taken from the UKMT Mathematical Challenges.
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