### Brain Teasers

# 5+2+1=8

Cryptography
In Cryptography teasers, a phrase or expressions has been encoded in some way (frequently by replacing letters with other letters). You need to figure out the encoding method and then decode the message to find the answer.

In the equation FIVE+TWO+ONE=EIGHT, each letter represents a different digit. Can you figure out which letter is which digit?

The numbers cannot begin with 0, i.e. none of F, T, O, and E can be 0.

Since the solution is not unique, a restriction is added: the product FIVE*TWO*ONE must be the greatest value possible.

The numbers cannot begin with 0, i.e. none of F, T, O, and E can be 0.

Since the solution is not unique, a restriction is added: the product FIVE*TWO*ONE must be the greatest value possible.

### Hint

Start from the first digits, then the last digits.### Answer

First, there are ten different letters in this equation: E, F, G, H, I, N, O, T, V, W. Therefore, every digit from 0 to 9 is used.Obviously E=1 and I=0. Hence IVE+TWO+ONE<2000, and F=9.

The remaining digits are 2, 3, 4, 5, 6, 7, 8. The ones digits read 1+O+1=T, hence T=O+2, and there's no carrying from the ones digits to the tens digits.

The tens digits read V+W+N=H. Because the letters V, W, N all appear only once in the equation at the tens digits, they are interchangeable. Since 2+3+4=9>8 and 6+7+8=21<22, the tens digits carry one to the hundreds digits. Hence V+W+N=10+H, or H=(V+W+N+H)/2-5.

Now we have FI+T+O+1=EIG, or 90+T+(T-2)+1=100+G, or G=2*T-11. From the digits 2 through 8 we can have T=7, G=3 or T=8, G=5.

If T=7, G=3, then O=5. But then we'll have H=(V+W+N+H)/2-5=(2+4+6+8)/2-5=5, which is occupied by O.

If T=8, G=5, then O=6, H=(V+W+N+H)/2-5=(2+3+4+7)/2-5=3, and V, W, N are 2, 4, 7 in some order.

Finally, since we want the largest product FIVE*TWO*ONE possible, and the sum FIVE+TWO+ONE=EIGHT=10538 is fixed, we need the least possible difference between them. Hence FIVE=9021, TWO=846, ONE=671.

We have 9021+846+671=10538 and 9021*846*671=5120914986, which is the greatest product possible.

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