Working Backwards through Chains

Some questions tell you how a story ends and ask how it began. I add fifteen, take away nine, and get forty-two — what did I start with? Draw the chain forwards as arrows: start, plus fifteen, minus nine, forty-two. Now walk backwards along the arrows. The last thing that happened was minus nine, so undo it: forty-two plus nine is fifty-one. Before that was plus fifteen, so undo it: fifty-one minus fifteen is thirty-six. Then check by running thirty-six forwards: plus fifteen is fifty-one, minus nine is forty-two. It lands where the story ended, so it is right.

Multiplying and dividing follow the same rule, and now the order really matters. I triple a number, add seven, and get thirty-four. Undo the last step, the plus seven: thirty-four minus seven is twenty-seven. Undo the tripling: twenty-seven divided by three is nine. If you had divided first you would be dividing thirty-four by three and getting a fraction, which is your signal that the order is wrong. Halving chains come up a lot: a number is halved three times and ends at seven — so double it three times going back: fourteen, twenty-eight, fifty-six.

Here is the shape the selective test uses. Alan had ten dollars. He gave some to Kiana. Half of what was left went to Jason, and Jason gave half of his to Lenny, who got one dollar sixty. What did Kiana get? Draw the chain: ten, minus Kiana's share, halve, halve, one sixty. Walk back: Lenny's one sixty doubled is Jason's three twenty; doubled again is six forty, which is what Alan had after Kiana. So Kiana got ten minus six forty, three dollars sixty. Notice the last step — the question asks for Kiana's share, and six forty is only the leftover. Stop when you have answered what was asked.

There is a second way to see the same story, and it is worth having both. Draw Alan's ten dollars as a bar. Kiana's share is one piece; what is left is the other. That leftover gets halved — Alan keeps half, Jason gets half — and Jason's half is halved again, so Lenny's piece is a quarter of the leftover. Lenny's quarter is one sixty, so the leftover is four times that: six forty. Kiana's piece is the rest of the ten. Arrows show the order of events; bars show the fractions. When both methods give three sixty, you can be sure.

Longer chains are only more steps of the same walk. Add twenty-five, halve, subtract seven, and get twenty. Back along the arrows: twenty plus seven is twenty-seven, doubled is fifty-four, minus twenty-five is twenty-nine. Then the check that makes this method safe: run twenty-nine forwards. Plus twenty-five is fifty-four, halved is twenty-seven, minus seven is twenty. It lands on the ending the question gave, so twenty-nine is right. The forwards check catches every mistake of order, and it takes ten seconds. Quiz and exercises now.