Letters to Numbers: What goes in the blank circle?

This is a letter to number conversion and pattern puzzle.

What number goes in the blank circle and why?

Each letter is a distinctly seperate digit from 0 to 9. ( There are 10 letters representating 10 seperate digits from 0 to 9).

All the vowel digits add up to a Prime number.

N^M = A

I^S = D

N! = C(T-O)^I

N = I^2

Please explain the logic

Attribution: Part of the puzzle inspired by a famous puzzle from the Orient which I will reveal after the answers are in

• Should [no-computers] be added? – Culver Kwan Jul 12 '20 at 13:04
• Finding the values for each letter is easy enough, but something in the diagram isn't adding up...? – Daniel Mathias Jul 12 '20 at 13:36
• Very well put @DanielMathias – DrD Jul 12 '20 at 13:37
• Are you sure the very bottom one shouldn't contain the letters $AM$ rather than $AA$? – Earlien Jul 12 '20 at 14:08
• Yes I am sure @Earlien – DrD Jul 12 '20 at 14:56

2 Answers

The final answer is

9

To figure out the digits (building off some of the logic in Earlien's solution):

$$N=I^2$$ means $$I$$=2 or 3 and $$N$$=4 or 9. 0 or 1 are not possible if $$N$$ and $$I$$ are to be unique.

Since $$N$$ = 4 or 9, $$N^M=A$$ means $$M$$=0 and $$A$$=1 ($$M$$=1 will result in non-uniqueness, and larger $$M$$ result in $$A$$>9).

$$I^S=D$$ means $${I,S,D}$$ = {3,2,9} or {2,3,8} since 0 and 1 are taken. However, $$I$$=3 and $$D$$=9 is impossible because $$N$$=9 when $$I$$=3. So now we can confirm $$I$$=2, $$N$$=4, $$D$$=8, and $$S$$=3.

$$N! = C(T-O)^I$$ simplifies to $$24/C = (T-O)^2$$. $$C$$=6 because that is the only possibility that gives a perfect square. Then $$T$$=9 or 7 and $$O$$=7 or 5.

Since the vowels must add up to a prime number, $$E$$=3, 5, or 9 if $$O$$=5, or $$E$$=1, 3, 7, or 9 if $$O$$=7. Since all those digits are taken except for $$E$$=9, then $$O$$=5, $$E$$=9, and $$T$$=7. We now have all the digits we need.

To figure out the missing number:

My first instinct was also to subtract the inputs from each other. However, this fails in the last step, where 20-10 != 11.

Instead, it seems you add all the digits of the inputs, so 1+3+2+3 = 9. Incredible how most of the numbers satisfy both patterns, until the last step befuddles you! Honestly, if I wasn't thinking about digits already, this would have been even tougher.

Upon some searching it seems this puzzle is inspired by:

Nob's number puzzle, a famous and similar number tree puzzle designed by Japanese puzzle guru Nobuyuki Yoshigahara.

Very clever, thanks for the puzzle!

• Very nice logical thinking @BrainEaser. Glad you liked it – DrD Jul 12 '20 at 21:49

Starting with $$N=I^2$$,

$$N$$ has to be either 4 or 9 (with $$I$$ being 2 or 3 respectively) otherwise $$N$$ and $$I$$ won't be unique (both 1) or too large. Given that there is a also an $$N!$$, it seems unlikely $$N$$ would be 9, so $$N=4$$ is a fairly safe assumption. And $$I=2$$.

Then

$$N^M=A$$ implies that $$M$$ must be zero and $$A$$ = 1 for the same reasons above - $$A$$ will be too large, or if $$M=1$$, then $$A=N$$ and are thus not unique.

From here,

it is pretty clear from $$I^S=D$$ must mean that $$S=3$$ and $$D=8$$ as there are no other small enough numbers for $$S$$ that work. $$N! = C(T-O)^I$$ simplifies to $$24/C = (T-O)^2$$. The only way that 24 will divide evenly with the remaining choices is if $$C=6$$, which means $$T-O=2$$. Thus $$T=7$$ and $$O=5$$. (As pointed out in the comments, at this stage, another possibility is $$T=9$$ and $$O=7$$, however, this will not satisify the last condition, discussed in the next step).

Which just leaves

$$E=9$$. And the vowels add up to 17 which is a prime number as advised.

For the second part, I do not know what the numbers (or equivalently, letters) should be in the missing circle.

• You need to eliminate the possibility of $T-O=9-7$ (easy to do) – Daniel Mathias Jul 12 '20 at 13:35
• In the first step above, there was another possibility as well, but making a few assumptions along the way is helpful and they are soon confirmed as correct. – Earlien Jul 12 '20 at 13:36
• I'm intrigued what the famous puzzle from the Orient is and how it relates to this one. – Earlien Jul 12 '20 at 13:59