Symbol Puzzles: Two Equations, Two Unknowns

Three stars and five hearts score twenty-seven; five stars and seven hearts score forty-one. Two unknowns, two facts. Compare them: the second row has two more stars and two more hearts, and fourteen more points. So two stars and two hearts are worth fourteen, and one star and one heart are worth seven. If the question asks for a star and a heart together, that is the answer already. If it wants each one: three stars and three hearts are twenty-one, so the other two hearts in the first row are worth six, a heart is three, and a star is four. Check both facts before you move on.

Sixteen medals scored thirty-six points, gold worth three, silver two, bronze one, and six of the medals were silver. Strip out what you know: six silvers are twelve points, leaving ten medals worth twenty-four. Now two unknowns, two facts: gold plus bronze is ten, and three times gold plus bronze is twenty-four. Compare: the second fact has two extra points for every gold, and fourteen extra points in total, so seven golds. Or think of it as all ten being bronze — ten points — and swapping bronze for gold two points at a time until you reach twenty-four: seven swaps. Check: seven golds, three bronzes, six silvers, thirty-six points.

When one fact tells you what a symbol equals, put that into the other fact. A heart is worth two more than a star, and a star and a heart are worth twelve. Replace the heart with star plus two: star plus star plus two is twelve, so two stars are ten, a star is five and a heart is seven. The same move handles 'Sarah ate twice as many as Tom, twelve altogether': Tom plus two Toms is twelve, so Tom is four and Sarah eight. There is also a quick way to see it: if the two were equal they would be six each, and moving one from the star to the heart makes the heart two more.

Three boxes weighed two at a time: A and B are twenty-eight, B and C thirty-two, A and C thirty-eight. Add all three weighings: ninety-eight. Every box was on the scale twice, so the three together weigh half of that, forty-nine. Then each box is the total minus the pair it was not in: A is forty-nine minus thirty-two, seventeen; B is forty-nine minus thirty-eight, eleven; C is forty-nine minus twenty-eight, twenty-one. Check every pair. If the question only wants the total, the halving trick answers it in one line.

When the comparing move is hard to see, guess and check is a perfectly good method. Four stars and two hearts make twenty-six, and a star and a heart together make eight. Try a star of three: then a heart is five, and four threes plus two fives is twenty-two, too small, so stars need to be bigger. Try six: heart is two, total twenty-eight, too big. Five: heart is three, total twenty-six. Each wrong guess tells you which way to move. On a multiple-choice paper the guesses are supplied — test the options. Quiz and exercises now.