# Brainteaser Answers – Issue 40, 2018

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## 1.) Loopy Lapel

**Solution:**

There are many YouTube videos that show the setup and solution, here is a good one: **https://www.youtube.com/watch?v=MrN1hNzXoec**

## 2) Algebradabra

Find all real numbers x such that:

**Solution:**

Square both sides (just remember that this process may give extraneous solutions – we will test each for validity later) to get:

Square again to get:

##### x^{4} – 10x^{2} + x + 2 = 0

Or:

##### (x^{2} + x – 5)(x^{2} – x – 4) = 0

Setting the second term (x^{2} – x – 4) to zero gives:

but neither answer satisfies the original equation.

Setting the first term (x^{2} + x -5) to zero yields:

but only

is a valid solution. And therefore it is the answer.

## 3) Alphametics

**Solution:**

Since each letter in the alphametic represents a unique and different “digit” and there are eight different letters used *(O, N, E, T, W, F, U, R), *the positive integral base must be equal to, or greater than, eight.

Some “truths” we can determine without regard for the base:

*O*+*O*=*R,*but*O*+*O*=*T;*therefore, in*ONE+ ONE= TWO, a*1 must be carried over from*N*+*N.*- Therefore
*2O = R, 2O + I = T, R + l = T, R*is even,*T*is odd, and*F*= I. - If
*O*= zero, then*R*= zero. Therefore*O*is neither 0 nor 1. - Also,
*O*must be less than one-half of the base, since 1 +*O*+*O*=*T*does not exceed the base, and*O*+*0*=*R*does not exceed the base.

Now let’s label the base *b.*

1 < *O *< b/2.

- If
*b*= 8,*O*= 2 or*O*= 3. (Neither works.) - If
*b*= 9,*O*= 2,*O*= 3, or*O*= 4. (None of these works.) - If
*b*= 10, then*O*= 2,*O*= 3, or*O*= 4. (None of these checks). - If
*b*= 11 or*b*= 12, then*O*= 2,*O*= 3,*O*= 4, or*O*= 5. (None of these checks in either base.) - If
*b*= 13,*O*= 2,*O*= 3,*O*= 4,*O*= 5, or*O*= 6. (None of these checks in base 13.) - If
*b*= 14,*O*= 2,*O*= 3,*O*= 4,*O*= 5, or*O*= 6.

If the digits of base 14 are {1, 2, 3, 4, 5, 6, 7, 8, 9, tn, el, tw, th}

Then when *O *= 4, a solution is found:

*ONE *+ *ONE *= *TWO *– 4(tn)2 + 4(tn)2 = 964,

*TWO *+ *TWO *= *FOUR *– 964 + 964 = 14(tw)8.

Therefore, *O *= 4, *N *= tn, *E *= 2, *T *= 9, *W *= 6, *F *= 1, *U *= tw, and *R *= 8.

**Thus, the smallest base for which these alphametics hold true is 14.**

[credit: *Mathematics Student Journal* – circa 1965]