6
$\begingroup$

I've created a mapping between words and complex-valued polynomials, and I've generated examples from word lists that I've I found on another puzzle. You don't need to study every one of these, but I recommend you study a few (1-5 of your choice) of them to get the hang of what the pattern is.

$$125741 x^{4} + 121879 i x^{3} - 39265 x^{2} - 4937 i x + 210 = \text{diet}$$ $$52327 x^{3} + 19518 i x^{2} - 1541 x - 30 i = \text{set}$$ $$1123441 x^{4} + 816565 i x^{3} - 161237 x^{2} - 10547 i x + 210 = \text{poem}$$ $$2430827 x^{4} + 1215483 i x^{3} - 184035 x^{2} - 10789 i x + 210 = \text{dust}$$ $$87857033 x^{5} + 121851510 i x^{4} - 37954862 x^{3} - 4128036 i x^{2} + 169961 x + 2310 i = \text{suite}$$ $$82786 x^{4} + 187003 i x^{3} - 100910 x^{2} - 10151 i x + 210 = \text{tape}$$ $$171039 x^{4} + 405755 i x^{3} - 208887 x^{2} - 13021 i x + 210 = \text{tube}$$ $$158097049 x^{5} + 202359278 i x^{4} - 48570918 x^{3} - 4478188 i x^{2} + 171809 x + 2310 i = \text{quote}$$ $$1622871789 x^{7} + 9191060836 i x^{6} - 13856459083 x^{5} - 7787386342 i x^{4} + 2084272163 x^{3} + 286658696 i x^{2} - 19453349 x - 510510 i = \text{disobey}$$ $$711942 x^{6} + 4455847 i x^{5} - 9967276 x^{4} - 9738036 i x^{3} + 4105552 x^{2} + 620917 i x - 30030 = \text{biased}$$ $$71489 x^{3} + 26306 i x^{2} - 1967 x - 30 i = \text{yes}$$ $$410451 x^{4} + 1045535 i x^{3} - 210247 x^{2} - 12557 i x + 210 = \text{tomb}$$ $$7854 x^{5} + 33715 i x^{4} - 55187 x^{3} - 43175 i x^{2} + 16159 x + 2310 i = \text{badge}$$ $$473316743 x^{5} + 218054962 i x^{4} - 38764626 x^{3} - 3342128 i x^{2} + 140431 x + 2310 i = \text{joust}$$ $$475887 x^{4} + 457703 i x^{3} - 106263 x^{2} - 8737 i x + 210 = \text{bump}$$ $$89815 x^{4} + 111121 i x^{3} - 42383 x^{2} - 5447 i x + 210 = \text{cite}$$

The question is, what polynomial corresponds to the word "mist"?

$\endgroup$
7
$\begingroup$

I think the answer is

$\text{mist} = 4485851 x^4 +1580965ix^3 - 200803 x^2 - 10859ix + 210$

Reasoning

Let $p_m$ represent the $m$th prime number (e.g, $p_1 = 2, p_2 = 3, \ldots$).
For each letter $l$ in a word $w$, let $f(l, w)$ be the index of the letter $l$ in $w$ (e.g, $f(d, diet) = 1, f(m, tomb) = 3$).
Furthermore, let $g(l)$ be the position of letter $l$ in the alphabet (e.g, $g(a) = 1, g(m) = 13$).

Then the formula for the polynomial corresponding to word $w$ is $$ P_w(x) = \displaystyle \prod_{l \in w} (p_{g(l)}x + p_{f(l,w)} i)$$

Example

$$P_{poem}(x) = (p_{g(p)}x + p_1 i)(p_{g(o)}x + p_2 i)(p_{g(e)}x + p_3 i)(p_{g(m)}x + p_4 i) $$ $$ = (p_{16}x + p_1 i)(p_{15}x + p_2 i)(p_{5}x + p_3 i)(p_{13}x + p_4 i) $$ $$ = (53x + 2i)(47x + 3i)(11x + 5i)(41x + 7i) $$ $$ = 1123441 x^4 + 816565 i x^3 - 161237 x^2 - 10547 i x + 210$$ Similarly,
$$ P_{mist}(x) = (p_{13}x + p_1i)(p_9 x + p_2i)(p_{19}x + p_3i)(p_{20}x + p_4i) $$ $$ = (41x+2i)(23x+3i)(67x+5i)(71x+7i) $$ $$ 4485851 x^4 + 1580965i x^3 - 200803 x^2 - 10859 i x + 210 $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.