Grade 4 Number Patterns

What this problem type covers

Fifteen problem types across five levels: a rule linking three numbers in a box, a growing line of calculations, a number table read up and to the left, two jumps taken in turn, adding a series by pairing its ends, halving paper repeatedly, a stack whose increase itself grows, matchstick shapes sharing sides, five calendar numbers in a shape, counting the digit pairs that satisfy an equation, a place in a triangular arrangement, a place across a run of cards, the digits of a long product, black stones counted from the white ones in a triangular arrangement, and a starred cell on a spiralling numbered board.

Example problems

One example from each difficulty band, with worked solutions.

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1. 1 · ★1
   Find the rule and work out the missing number.
   

   • 18 | 10 → 19
   • 19 | 9 → 19
   • 58 | 13 → 26
   • 30 | 14 → 17

   If the two numbers above are 42 and 11, what goes below?
   • ①13
   • ②17
   • ③53
   • ④44
   • ⑤18
   Show solution

   ②17

   1. Compare the number below with the two above it.
   2. The digits of 42 add to 6.
   3. Adding that to 11 gives the number below.
   4. So the answer is 17.
2. 2 · ★2
   Part of a number table. Find the rule and give the number 1 up and 1 left of the top-left entry.
   

   • 46,284　46,294　46,304　46,314
   • 47,284　47,294　47,304　47,314
   • 48,284　48,294　48,304　48,314
   • ①45,274
   • ②45,284
   • ③46,284
   • ④45,294
   • ⑤47,274
   Show solution

   ①45,274

   1. See how much the table gains going right and going down.
   2. Down is 1,000 a step, right is 10 a step.
   3. Going up subtracts, and so does going left.
   4. The number is 45,274.
3. 3 · ★3
   A sheet is cut in half, the pieces are stacked and cut in half again, and so on. After 8 cuts, how many pieces are there?
   • ①16
   • ②258
   • ③512
   • ④128
   • ⑤256
   Show solution

   ⑤256

   1. Each cut doubles the number of pieces.
   2. One cut gives 2, two cuts give 4.
   3. 8 cuts multiply by 2 that many times.
   4. That is 256 pieces.
4. 4 · ★4
   How many pairs of digits from 1 to 9 satisfy this?
   30 − A = 23 − B
   • ①3
   • ②1
   • ③4
   • ④7
   • ⑤2
   Show solution

   ⑤2

   1. Set the two sides equal and compare.
   2. The left is 7 more, so A must be 7 more than B.
   3. Count every such pair from 1 to 9.
   4. There are 2.
5. 5 · ★5
   Stones are laid out by a rule. When 29 white stones have been placed, how many black ones are there?
   
   • ①169
   • ②29
   • ③91
   • ④104
   • ⑤13
   Show solution

   ③91

   1. The white stones are the bottom row and the first stone of every other row.
   2. So step n has (n+1)+(n+2) white stones, and that count gives n = 13.
   3. The black stones go 1, then 1+2, then 1+2+3, and so on.
   4. Step 13 therefore has 1+2+…+13 = 91 black stones.

Where people usually go wrong

• Adding the two numbers above instead of using the digit sum of one of them.
• Taking an alternating pattern as a single repeated jump.
• Halving after multiplying the pair total by the wrong count.
• Treating a growing increase as a constant one.
• Charging a full set of matchsticks for every shape; joined shapes share a side.
• Answering with the middle calendar number when the smallest was asked for.