Concepts · Number series

Interleaved series

When a series jumps up and down, it can be two series taking turns.

Kabir has the difference table now, and he reaches for it:

2, 10, 4, 15, 6, 20, ?

Gaps: +8, −6, +11, −9, +14. Up, down, up, down, and no two the same. The gaps of the gaps are worse. He decides the table is broken and picks 25, because it is bigger than 20.

The carrom score sheet

That evening he plays carrom with Riya and keeps the score himself, in one line on a scrap of paper, a number after every turn: 5, 1, 10, 3, 15, 5.

“Who’s winning?” asks Riya.

Kabir reads every other number. “You: 5, 10, 15. Me: 1, 3, 5.” Then he looks at his own scrap of paper again. One line, two players taking turns. And whose turn is next? The seventh, an odd turn: Riya’s.

A turn-by-turn score sheet is not one list. You read it player by player, and you keep track of whose turn it is. Kabir takes out the mock paper.

The rule

When no single step fits a series, and especially when its gaps jump up and down, first check for one rule that alternates (× 2, then − 3, then × 2 again). If there is none, read the terms in odd places (1st, 3rd, 5th…) as one series and the terms in even places as another. Solve each one. Then count which place the blank is in, and continue that series.

The places are places in the series: the 1st term, the 2nd term, and so on. They have nothing to do with positions in the alphabet.

Kabir’s question, again

1. Odd places: 2, 4, 6, going up by 2. Even places: 10, 15, 20, going up by 5.
2. The blank is the seventh term. Seven is odd, so it belongs to 2, 4, 6.
3. The answer is 8.

And 25, the number Kabir picked? It is the right next number for the other strand. It would come one turn later.

Letters take turns too

A, Z, C, X, E, V, ? Odd places climb from A: A, C, E. Even places fall from Z: Z, X, V. The blank is the seventh place, and seven is odd, so it is the next climbing letter: G.

Why it helps in the paper

Here each strand on its own is easy; braided together, they are not. The trap to watch for in the options is the right number from the wrong strand, the one that would come a turn later. Counting the blank’s place takes a moment and removes the trap.

A strand can be a simple adding series: The difference table.

Try three

Next mock

A week later: 7, 2, 14, 4, 21, 8, ? Kabir sees the jumps, checks for one alternating rule, finds none, and counts the places. The blank is the seventh, an odd place: 7, 14, 21, 28. He writes 28, and does not even look at the 16 in the options.