Logic Level 4

$$\large\begin{array} &&C&H&A&R \\ \quad &&R&A&Y\\ + \quad S&A&U&R&A\\ \hline \qquad E&M&B&E&R \end{array}$$

TL;DR

1)Someone told you that they found $$2$$ solutions to the cryptogram above and that the two $$\overline{RAY}$$ of their solutions sum up to $$1256$$

2)Upon hearing this and doing some calculations, you claim you can do better, by finding $$4$$ solutions to the cryptogram above. The $$\overline{RAY}$$ of your two additional solutions also sum up to $$1256$$.

3)Justify your claim and enter the sum of all $$4$$ possible values of $$\overline{CHAR}$$ into the answer box.

"Guys, we have rescue missions to run. We can't stick around wasting our time solving this puzzle. There might be no solutions at all," Char the charmander lamented, cupping his tail flame with both hands. His tail flame has gotten pretty large, evident of his frustration, which he was trying to hide from his team.

"To tell you a fact, I spotted two solutions within the first minute. To think that the three brains of your team can't match the one of an old bug..." Scythe sighed.

"Can I get a hint, please?" Ray the raichu pleaded. "I promise, I will make sure we solve this cryptogram before dinner. We will even skip dinner to work on it if necessary."

"RAY!" Char and Saura exclaimed in unison.

"Heh, frivolous optimism. I like your attitude, Ray. Let me give you a hint, then. For the two sets of solution I found, the two $$\overline{RAY}$$s add up to $$1256$$. I expect your teammates to hold up the end of your bargain." Scythe said with a mischievous grin.

With blinding speed, the scyther zipped out of the door. Before the members of Team Ember had time to even bat their eyelashes, the sound of a clicking padlock was audible. They had been locked in! In their own room, no less.

"You have three chances. Tell me the answer and you will be free to go for dinner!" Scythe's voice seemed awfully cheerful.

"What... RAY!! Whaaat have you dooone!" Char yelled, practically mourning for the loss of a meal.

Suddenly, an idea clicked in Ray's mind. "Guys, I think I know how to solve this. Give me a moment!"

With a piece of charcoal in his hand, Ray furiously worked out the solution, only to be disappointed less than a minute later.

"Guys, Sycthe isn't making this easy for us. I tried the most straightfoward approach, thinking it would work, but it wouldn't." Ray slumped over onto the floor, still reeling in disappointment.

"However, this means that we can now find $$4$$ solutions to this cryptogram. Aren't you guys excited? Come on, remember our team motto? The fire will never die, until the last ember fades!" Ray said, trying to put on a cheerful mood.

Both Char and Saura pretended to have not heard Ray. They were still bitter about missing dinner.

"Hey, Scythe! Based on your hint, I promise that I will be able to find four sets of solutions. If necessary, we will skip tomorrow's breakfast to work on this." Ray spoke excitedly.

"You are a clever little rodent, aren't ya?" Sycthe laughed. "Four solutions, gotcha."

Char's and Saura's eyes just glossed over.

$$1)$$ Explain Ray's change in mood, from his initial excitement to disappointment to his later bold claim that four distinct solutions can be found. (Mathematically, of course!)

$$2)$$ Enter the sum of all $$4$$ possible values of $$\overline{CHAR}$$ into the answer box.

Reference: Silver Resistance fanfiction

In the fanfiction, Char the Charmander, Saura the Bulbasaur and Ray the Raichu are the founding members of Team Ember, a resistance team opposing the evil forces of the Master.

This question is part of the problem set Mathematics in Anime

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