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.
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
This problem is taken from the UKMT Mathematical Challenges.