122 practice questions covering Number Play, organised into 0 topics. 122 questions include a written explanation. Pick a topic below, or try the samples first.
Which number appears in the Collatz sequence starting from $6$?
Correct answer: B - $10$
The chain runs $6 \to 3 \to 10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1$, so $10$ appears. The numbers $12$, $9$ and $7$ never occur in this chain.
A shop sells $146$ pens on Monday and $189$ on Tuesday. How many pens in all?
Correct answer: C - $335$
Adding the hundreds and tens first, $146 + 189 = 146 + 200 - 11 = 335$. Answering $325$ or $345$ is out by ten, and $330$ is out by five.
Starting from $1234$, how many Kaprekar steps are needed to reach $6174$?
Correct answer: D - $3$
The chain runs $4321 - 1234 = 3087$, then $8730 - 0378 = 8352$, then $8532 - 2358 = 6174$, so $3$ steps are needed. One or two steps stop short at $3087$ or $8352$, and the constant is reached before a fifth step.
What is $63 + 28$?
Correct answer: D - $91$
Breaking it up, $63 + 20 = 83$ and $83 + 8 = 91$. Answering $81$ forgets a ten, $85$ mishandles the units, and $92$ overshoots by one.
In the same game, $10$ sticks remain and it is your turn. How many should you take to be sure of winning?
Correct answer: D - $2$
Taking $2$ leaves $8$, a multiple of $4$ and therefore a losing position for your opponent. Taking $1$ leaves $9$ and taking $3$ leaves $7$, both of which let your opponent move to $8$ instead, so the choice certainly matters.
In the take-$1$-to-$3$ game where the last stick wins, which of these is the smallest losing position for the player about to move?
Correct answer: D - $4$
With $1$, $2$ or $3$ sticks the player to move simply takes them all and wins, so those are winning positions. With $4$, every move leaves a pile the opponent can clear, making $4$ the smallest losing position.
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