## The Hidden Beauty

It is rare when a word equation coincides with a number equation.

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Problem.A store sells letter magnets. The same letters cost the same and different letters might not cost the same. The word ONE costs 1 dollar, the word TWO costs 2 dollars, and the word ELEVEN costs 11 dollars. What is the cost of TWELVE?

## James:

Let x = cost of “O”.

Let y = cost of “NE”.

Let w = cost of “TW”.

Let z = cost of “ELEV”.

Then we have:

x + y = 1

w + x = 2

y + z = 11

From the first two equations, we have w – y = 1. Subbing for y, you get w – (11 – z) = 1 w + z = 12. Noting that w = “TW” and z = “ELEV” or “ELVE”, then w + z = “TWELVE”. So the cost of “TWELVE” is 12 dollars.

29 March 2017, 3:55 pm## Dave:

“TWELVE ONE” is an anagram of “TWO ELEVEN” so the value of “TWELVE” is the sum of the values of “TWO” and “ELEVEN” less the value of “ONE” (or 2+11-1=12)

29 March 2017, 8:36 pm## tanyakh:

Great!

That was my point: TWELVE + ONE = TWO + ELEVEN in letters, as well as the corresponding numbers.

30 March 2017, 4:26 pm## tanyakh:

I received a comment by pupugai that a similar equation exists in Russian. It includes numbers 5, 6, 15, 16.

30 March 2017, 4:27 pm## Al Yunani:

In Greek also Tanya. 5=ΠΕΝΤΕ, 6=ΕΞΙ, 15=ΔΕΚΑΠΕΝΤΕ (10=ΔΕΚΑ), thus ΔΕΚΑΕΞΙ (pronounced “deka eksi” and meaning sixteen )=16.

1 April 2017, 6:59 amBut, since no anagram is involved…rather trivial I would say. 🙂

## Αγγελος:

It does, but trivially so. 5 is пять, 15 is пятнадцать, 6 is шесть and 16 is шестнадцать.

1 April 2017, 7:00 am## Al Yunani:

But, also the Russian seems pretty “trivial”. Right?

1 April 2017, 7:10 am5 = пять, 15 = пятнадцать, 6 = шесть and 16 = шестнадцать.

## Ted R:

My gosh, when did Ls get so expensive? When I was a kid you could get 3 Ls and a pack of gum, all for a buck fifty.

4 April 2017, 1:44 pm## Ted R:

PS: And I suppose that NONE would be free?

4 April 2017, 2:47 pm## Ted R:

I guess my first joke backfired – it turns out that Ls are free! I had to dig around to find a few more equations, but if

NONE = 0

LOVE = 0 (a la tennis)

ONE = 1

LONE = 1

TWO = 2

TEN = 10

ELEVEN = 11

then (E, L, N, O, T, V, W) = ( 14/3, 0, -1, -8/3, 19/3, -2, -5/3 )

Case closed (& morning wasted).

4 April 2017, 4:39 pm## tanyakh:

Ted R,

This is beautiful. (Your morning wasn’t wasted.)

5 April 2017, 3:45 pm