Number Play: questions with answers

Only sometimes. The rule looks only at the children IN FRONT. Anybody with nobody taller ahead of them says 0. The child standing first says 0 whatever their height, because there is nobody ahead at all. So a short child near the front can call 0 as easily as the tallest child can.

A call of 0 means no taller child stands in front. Since all heights are different, everyone in front is shorter. The call tells us nothing about the heights of children behind. One of them could be taller, so the caller need not be tallest.

144. Each term is the sum of the two before it, so the next one is 55 + 89 = 144. Doubling would give 178, which fits the opening 1, 2 by accident and nothing after it -- the third term is 3, not 4.

21, 34, 55. Each one is the sum of the two before it, so the sequence is built up a step at a time from 8 and 13. Doubling each time would give 26, 52, 104, and adding the same amount each time would give 18, 23, 28: over three steps the three routes separate completely.

Both results are odd. Reversing a subtraction changes its sign but not its parity. Each result is the difference of an even number and an odd number. A negative odd number is still odd.

84 is even, so all the marbles can be put into pairs. An odd number is some pairs and one more. Taking that away leaves one marble without a pair. So the number left must be odd. Of the four numbers, only 77 is odd. For example, giving away 7 marbles leaves 84 - 7 = 77.

The number of rows is odd. The columns can be paired, so all small squares form pairs. So the total is even. The exact product is not needed to decide its parity.

Odd. A product is even only when one of its factors is even, and here none of them is, so nothing can supply a pairing. The COUNT makes no difference at all -- which is where addition misleads, because there an even count of odd numbers really does give an even total.

No -- the sum is even however many are added. Each even number is made of pairs with nothing left over. Putting these collections together still leaves nothing over. The sum is even whether two or twenty even numbers are added. The count changes the parity of a sum of odd numbers, but not of even numbers.

Odd. The even number is made of pairs with nothing left over, and the odd number brings one leftover with it. There is nothing for that leftover to pair with, so the total cannot be arranged in pairs. Here 78 + 53 = 131, which is odd -- but the answer does not depend on these two numbers at all.

Putting q = 1, 2, 3 in 4q - 2 gives 2, 6 and 10. Each value is 4 more than the one before, so the values go up in steps of 4. Every one of them is even, but the even numbers in between are stepped over. An expression that is always even need not give every even number.

90. The three row totals between them add every entry of the grid exactly once, which is 1 + 2 + ... + 9 = 45; the three column totals do the same and come to 45 as well. Together the six count every entry twice, so they add to 45 + 45 = 90. The 45 is the total of ONE direction, not of all six.

A row contains three different numbers from 1 to 9, so its total is between 6 and 24. The three row totals must also add to 45. Only the set 12, 15 and 18 passes both checks here. The rows 1, 2, 9 then 3, 4, 8 then 5, 6, 7 show that these totals can be made.

33. The grid is still magic, because every line has changed in the same way. But each line holds THREE entries, so each line gains 4 three times over -- that is 12 -- and the magic sum goes from 21 to 21 + 12 = 33. It neither stays put nor gains 4 just once.

S = 5 and R = 1. Three copies of S must end in S again, so twice S has to end in 0 -- which leaves S as 0 or 5. Zero gives a total of 0, a single digit, so S is 5, the total is 15 and R is 1. The chapter tries 2 and 3 and both fail on the units digit: 6 is not 2 and 9 is not 3.