Revision and important questions: Algebra Play

5.1 The formula box

• Trick steps: follow the starting letter through every step; the final expression shows the hidden constant.
• Two-row pyramid: top = a + b. Three-row: top = a + 2b + c. Four-row: top = a + 3b + 3c + d.
• Calendar 2 × 2 block: sum = 4a + 16.
• Largest product of three digits p < q < r: multiplier r, multiplicand qp, so r × qp.
• Reversed two-digit difference: 9(b − a); reversed sum: 11(a + b).
• Cyclic three-digit sum: 111(a + b + c) = 3 × 37 × (a + b + c).
• Repeating pattern: abcabc = 1001 × abc = 7 × 11 × 13 × abc.

5.2 Reading the formula box

The pyramid formulas are short because the same middle entries are reused at every level. Write the first few levels once and the coefficients become obvious; the ends enter once, and each step inward adds one more use. The calendar formula follows from the row and column offsets: moving right adds 1 and moving down adds 7. The largest-product rule is a comparison of expansions, not a lucky guess: after grouping by the multiplier, only the second terms differ. The divisibility identities all come from writing the number in place value, which is why the proofs are one or two lines each. When a trick is new, test it on two or three numbers before believing the general claim, then write the algebra to prove it; the examples show what to prove and the algebra shows why it is always true.

Two useful checks: substitute a small set of digits into the formula and compare with the answer computed by hand, and confirm that the coefficient sum of a four-row formula equals the top row when all four entries are set to 1.

5.3 Common mistakes

1. Forgetting the hidden constant at the end of a trick and thinking the trick depends on the starting number.
2. In a pyramid, adding upward in the wrong order or using the same middle number only once.
3. Mixing the coefficients of a four-row pyramid. Write the rows out once instead of memorising the pattern wrongly.
4. In the date trick, subtracting the wrong constant or forgetting that the day must be a two-digit block.
5. In the largest-product puzzle, trying all six products instead of grouping by the multiplier.
6. In divisibility proofs, writing an example as proof. An example checks a claim; the identity proves it.
7. In the reversed-number tricks, swapping a and b and losing the sign of the difference.

5.4 Ten exam-style questions with answers

Work through each question with pen and paper before reading the answer; the answers are short on purpose.

1. A trick says: double the number, add 6, divide by 2, subtract the original number. What is the answer? Answer: (2x + 6)/2 − x = x + 3 − x = 3.
2. Find the top of a three-row pyramid with bottom row 6, 9 and 4. Answer: 6 + 18 + 4 = 28.
3. Find the top of a four-row pyramid with bottom row 3, 4, 5, 6. Answer: 3 + 12 + 15 + 6 = 36.
4. A 2 × 2 calendar block has sum 48. Find its top-left number. Answer: 4a + 16 = 48, so a = 8.
5. Find the largest product from digits 4, 6 and 8. Answer: multiplier 8, multiplicand 64, so 8 × 64 = 512.
6. A two-digit number and its reversal add to 143. What can you say about the digits? Answer: 11(a + b) = 143, so a + b = 13.
7. Find the quotient when the number 85 is reversed and the difference is divided by 9. Answer: 85 − 58 = 27, and 27 ÷ 9 = 3 = 8 − 5.
8. Show that abcabc is divisible by 7, 11 and 13. Answer: abcabc = 1001 × abc, and 1001 = 7 × 11 × 13.
9. Karim starts with 15 coins, doubles and pays 8 each round for 3 rounds. How many coins remain? Answer: 8 × 15 − 8 × 7 = 120 − 56 = 64 coins.
10. A three-row pyramid has top 47 and bottom row 5, 9 and n. Find n. Answer: the top is 5 + 2 × 9 + n = 23 + n, so n = 24.

5.5 Exam strategy

For a trick, write the steps with x and simplify only at the end. For a pyramid, write the formula once and substitute. For a calendar block, label the top-left cell a and add the offsets before solving. For divisibility, express the number in place value and factor the result. For word problems, name each unknown, write one equation per sentence, and check the answer against the original wording.

Keep the answer in letters until the last line, then substitute only when numbers are given; this keeps the working short and makes the cancellation visible. For a word problem, check that the final value is sensible, for example a count of animals must be a whole number and cannot be negative.