Number Play

Think of time, dates, counting, measuring and money.

An end child has only one neighbour.

Stand them in a line with all children the same height.

Choose an arrangement and count each taller neighbour separately.

Place them from shortest to tallest.

The first child sees one taller neighbour, each middle child has one taller neighbour, and the tallest at the end has none.

The tallest child has no taller neighbour, wherever they stand.

Choose heights in this order: 5, 4, 1, 2, 3.

Their taller-neighbour counts are 0, 1, 2, 1, 0.

Put low children between taller children: high, low, high, low, high.

The two interior low children each have two taller neighbours.

Compare each number with its immediate left and right neighbours.

An end number has only one neighbour.

Place small values beside each required high cell.

Use a larger neighbour where a cell must fail to be a supercell.

Alternate a high value with a low value, starting and ending high.

Check each position against the number next to it.

Two neighbouring cells cannot both be strict peaks.

Alternate peak and non-peak positions, beginning with a peak.

The largest number in the row is greater than its only neighbour (if at an end) or both neighbours (if inside).

The largest exceeds all neighbours.

With distinct numbers, a smallest cell has a larger neighbour.

Put the largest number beside the second-largest.

Place the second-smallest at an end next to the smallest.

Place the largest next to the second-largest.

Change the row length, the number of peaks, or whether numbers may repeat.

Choose permutations of 10,639 for the empty cells.

Compare each cell with its neighbours that share a side.

Compare the ten-thousands and then following digits.

An even number ends in 0 or another even digit.

Compare the even entries.

Ignore entries below 50,000, then compare the remaining values.

Sort the values from least to greatest.

The labelled marks are evenly spaced. Count each equal step.

The labelled marks are evenly spaced. Count each equal step.

The labelled marks are evenly spaced. Count each equal step.

The labelled marks are evenly spaced. Count each equal step.

For k digits, the count is 9 followed by k−1 zeroes.

Add the digits to check each example.

A one-digit number cannot sum to 14.

For a two-digit number, make the tens digit as small as possible while the units digit is at most 9.

Put as much of the sum as possible in the leftmost digit: 9.

The next digit is 5; the rest are zero.

Insert as many zeroes as desired between 9 and 5.

For 40–49, the sums are 4 through 13.

For each new ten, restart from the tens digit and increase by one each number.

123 sums to 6; 234 to 9; each next digit-sum rises by 3.

From 1–100, count 7 in the ones place 10 times and in the tens place 10 times.

From 000–999, each place contains 7 in 100 numbers.

The first and last digits must match.

Choose the repeated digit and then any middle digit.

Example: 47+74=121.

Continue reversing and adding if the result is not yet a palindrome.

The units digit must be odd and small enough that twice it, then twice that, remain digits.

Units 1 gives tens 2 and hundreds 4; mirror the outside digits.

Arrange the digits greatest-to-least and least-to-greatest, using zeroes if needed, then subtract.

Look for repeated minute digits or reversed hour/minute digits.

Keep day 20, month 12 and use a year ending 20.

Read DDMMYYYY as one eight-digit string.

The weekdays shift by one after an ordinary year and two after a leap year.

A later year with the same leap status and January 1 weekday has the same calendar.

Use digits with a large spread between high and low place values.

Use digits closer together.

Choose digits that make a large descending number.

Choose a set with smaller place values.

Smallest=10,001; largest=99,999.

Add and subtract.

The next times are 11:11 and 12:21.

10:01→11:11 is 70 minutes; 10:01→12:21 is 140 minutes.

5683→5085→7992→7173→6354→3087→8352→6174.

Repeatedly add the available values for each target.

The only available value below 1,000 is 400, and repeated 400s do not total 1,000.

45,000=40,000+12,000−7,000.

5,900=7,000−800−300.

17,500=12,000+7,000−1,500.

21,400=12,000+7,000+3×800.

Build examples from the smallest or largest numbers in each digit class.

Maximum 4-digit sum=9,999+9,999=19,998.

Minimum 5-digit sum=20,000.

Maximum difference of two 5-digit numbers=99,999−10,000=89,999.

Test the smallest and largest possible inputs for each statement.

(a) There are twelve 40s and ten 50s: 12×40+10×50=980.

(b) There are 44 plain cells and 20 cells with five dots: 44+20×5=144.

(c) 32 cells show 32 and 16 show 64: 32×32+16×64=2,048.

(d) There are 18 purple cells with 3 dots and 12 red cells with 4 dots: 18×3+12×4=102.

(e) and (f) Multiply and add each printed label by its count.

If n is even, replace it with n÷2.

If n is odd, replace it with 3n+1, then repeat.

Apply half to even terms and 3n+1 to odd terms.

For steps, count a short sample and scale by distance.

For blinks or breaths, count for one minute, then multiply for longer periods.

Choose countable groups for a few thousand and larger populations or codes for more.

Estimate words per page and multiply by the number of pages.

Count or estimate students on a typical bus and the number of buses.

Prices and quantities vary.

A small amount of each inexpensive fruit may fit the budget; premium fruit or larger portions may not.

Use an India map scale or compare the cities’ positions.

At about 6 school hours per day and 200 days per year, that is 1,200 hours per year.

Over about eight years, that is roughly 9,600 hours.

Estimate distance and divide by a normal walking speed.

Use different scales for a short local trip and very long routes.

Choose a quantity that can be estimated by sampling or rounding.

Work backward from 21 by groups of 4.

The target numbers are 1, 5, 9, 13, 17, 21.

The combined turn total is 11.

Work back from 99 in steps of 11.

Choose the largest allowed addition m and target T.

Work backward in groups of m+1.

Swap 1 and 6 inside 62,871 to make 12,876.

Compare the resulting grid cells with their side-neighbours.

Arrange the four digits descending and ascending each round and subtract.

Continue until 6174 appears.

For the largest, maximise digits without exceeding 75,000.

For the smallest, minimise each digit while staying above 35,000.

For closest to 50,000, choose the least possible value above 50,000.

Count weekends, school vacations and holidays in your own calendar.

Avoid counting a holiday twice if it overlaps a weekend.

Use a measuring jug for a sample or compare with a known litre container.

18,000+300+370=18,670.

For 250, use ten groups of 25.

A power of 2 is even until it reaches 2.

Repeated halving gives 2→1.

Apply the rule successively, halving even numbers and using 3n+1 for odd numbers.

The target sequence is 2, 6, 10, 14, 18, 22.

The first player can say 2, then complement each opponent’s move to total 4.