T R Y
x A N E
A G Y Y
A E G S
B A F Y
B E F F Y Y
As, Y + S = Y
Hence value of S=0 Detailed Explanation-Rule 1
Put S=0 in main problem
T R Y
x A N E
A G Y Y
A E G 0
B A F Y
B E F F Y Y
Now you can easily predict the value of Y=5
[You will only get Zero at unit digit when you multiply 5 with an even number]
Possible ways of getting O at Unit Digit
Hence put Y=5 and rewrite the problem
T R 5
x A N E
A G 5 5
A E G 0
B A F 5
B E F F 5 5
Now proceed further..
[If you are sure about any variable, replace the variable with its digit and write the problem again. It will help you in collecting the more clues.]

1. A
2. D
3. C
T=4, R=6, Y=5, A=3, N=8, E=7, S=0, G=2, B=1, F=9
4 6 5
x 3 8 7
3 2 5 5
3 7 2 0
1 3 9 5
1 7 9 9 5 5

T R Y
x A N E
A G Y Y
A E G S
B A F Y
B E F F Y Y
As, Y + S = Y
Hence value of S=0 Detailed Explanation-Rule 1
Put S=0 in main problem
T R Y
x A N E
A G Y Y
A E G 0
B A F Y
B E F F Y Y
Now you can easily predict the value of Y=5
[You will only get Zero at unit digit when you multiply 5 with an even number]
Possible ways of getting O at Unit Digit
Hence put Y=5 and rewrite the problem
T R 5
x A N E
A G 5 5
A E G 0
B A F 5
B E F F 5 5
Now Possible values of A={3, 7, 9} and P={3, 7, 9}
Here you have one more clue i.e.
T R 5
x A N E
A G 5 5
A E G 0
B A F 5
B E F F 5 5
A + A + 1(carry)=E
Now you have to start hit and trial with the possible values of A={3, 7, 9} and E={3, 7, 9}
Firstly taking A=3 then E=7
put these values in main problem and write it again.
T R 5
x 3 N 7
3 G 5 5
3 E G 0
B 3 F 5
B 7 F F 5 5
Here you can easily predict the value of R=6
T R 5
x 7
3 G 5 5
i.e. 7 x 6 + 3 (carry)=_5(45)
Hence Put R=6 and rewrite it again
T 6 5
x 3 N 7
3 G 5 5
3 E G 0
B 3 F 5
B 7 F F 5 5
Now you can easily predict the other values
F=9, T=4, G=2, B=1
Hence,
4 6 5
x 3 8 7
3 2 5 5
3 E 2 0
1 3 9 5
1 7 9 9 5 5
[Relax - it's going to take some time to understand the whole concept. If you are facing any difficulty. Please go through the Cryptarithmetic Tutorial.]

4 6 5

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