MathsCBSE Class 8 · Algebra Play

Think-of-a-Number Tricks, Number Pyramids and Calendar Grids for Class 8

Q: Can the date trick work for any day?

Yes, because the day is at most 31, which has two digits; the trick is designed so the day is placed in the last two digits.

Q: How do I design my own calendar trick?

Choose a grid shape, write each date in terms of the top-left a (adding 1 across and 7 down), find the sum as an expression and solve for a when told the sum.

Q: Do number pyramids always have a unique solution?

If the bottom row is fully known, yes. With unknowns, you need enough equations to find them all.

Q: What is the Virahanka-Fibonacci sequence?

The sequence 1, 2, 3, 5, 8, ... in which each number is the sum of the two before it.

Updated 17 October 2026 · Fact-checked

Algebra explains number tricks: name the chosen number x, follow the steps symbolically and see that x cancels, for example 2x + 4, divided by 2 and minus x is always 2. In a number pyramid each number is the sum of the two below it, so a 3-row top is a + 2b + c and a 4-row top is a + 3b + 3c + d. In a calendar 2 x 2 grid with top-left a the sum is 4a + 16.

Follow the teal x blocks: doubling, adding 4, halving and subtracting x leaves just 2 small squares, so the answer never depends on x.

Illustration drawn for QuestionBuddy.

Understand Think-of-a-Number Tricks, Number Pyramids and Calendar Grids

Think-of-a-number tricks. Let the number be $x$ and apply the steps with algebra.
1. Think of a number: $x$ . 2. Double it: $2 x$ . 3. Add four: $2 x + 4$ . 4. Divide by two: $x + 2$ . 5. Subtract the original number: $x + 2 - x = 2$ .
The result is always 2, whatever $x$ is, because the letter-number cancels. To change the final answer to 3, add 6 in step 3 instead (and the division gives $x + 3$ ). Any sequence of steps where the coefficient of $x$ ends at 0 gives a fixed result.

The date trick. Let the month be $M$ and the day $D$ . The steps: multiply $M$ by 5, add 6, multiply by 4, add 9, multiply by 5, add $D$ . Algebraically: $5 M \to 5 M + 6 \to 20 M + 24 \to 20 M + 33 \to 100 M + 165 \to 100 M + 165 + D$ . Subtracting 165 leaves $100 M + D$ . Since the day is at most 31 (two digits), the last two digits are $D$ and what stands before them is $M$ . For an answer of 1390, subtracting 165 gives 1225: 25th December. The trick works because $100 M + D$ reads off the month and day directly; other trick designs would subtract a different number.

Number pyramids. In a number pyramid each number is the sum of the two numbers directly below it. Use letters for unknown cells and form equations. For a 3-row pyramid with bottom row $a, b, c$ , the middle row is $a + b$ and $b + c$ and the top is $a + 2 b + c$ . For a 4-row pyramid with bottom row $a, b, c, d$ the top is $a + 3 b + 3 c + d$ (the coefficients are 1, 3, 3, 1). So you can find the top without building the whole pyramid. The Virahanka-Fibonacci sequence $1, 2, 3, 5, \dots$ (each number is the sum of the two before it) gives interesting pyramids.

Calendar grids. In a calendar, the number below a date is 7 more. In a 2 x 2 grid with top-left number $a$ the numbers are $a, a + 1, a + 7, a + 8$ and their sum is $4 a + 16$ . If the sum is 36, then $4 a + 16 = 36$ , $a = 5$ . For a 3 x 3 grid with top-left $a$ , the sum is $9 a + 72$ , which is 9 times the middle number $a + 8$ . You can create your own trick with a grid of a different size or shape.

Grid puzzles. In a grid where shapes stand for numbers and the last column of each row is the sum of the values to its left, write equations for each row and solve.

The genie puzzle. Karim's coins double each round and he pays the genie a fixed number of coins each round. With $x$ coins and a payment $c$ per round, after one round he has $2 x - c$ , after two $4 x - 3 c$ and after three $8 x - 7 c$ . If $c < x$ he gains coins each round; if $c = x$ the number stays $x$ ; if the genie sets $c = \frac{8 x}{7}$ , the genie takes everything after 3 rounds.

The largest product and divisibility tricks are on the next topic page of this chapter.

Think-of-a-Number Tricks, Number Pyramids and Calendar Grids: formulas to remember

Think-of-a-number cancellation: $\frac{2 x + 4}{2} - x = 2$; The letter-number cancels, so the result is fixed.
Date trick: $5 (4 (5 M + 6) + 9) + D = 100 M + D + 165$; Subtract 165 to get 100M + D.
Number pyramid, 3 rows: $top = a + 2 b + c$; Bottom row a, b, c.
Number pyramid, 4 rows: $top = a + 3 b + 3 c + d$; Bottom row a, b, c, d.
Calendar grids: $2 \times 2 : 4 a + 16; 3 \times 3 : 9 a + 72$; a is the top-left number; the dates in a row differ by 1 and in a column by 7.
Genie puzzle after n rounds: $2^{n} x - (2^{n} - 1) c$; x coins at the start and c coins paid per round.

How to solve Think-of-a-Number Tricks, Number Pyramids and Calendar Grids questions

Use a letter-number for the unknown and translate each step.

1Choose a letter-number for the starting value (x, or M and D for a date).
2Apply each instruction as an operation on the expression and simplify after each step.
3If the letter-number cancels, the result is fixed; if not, the result tells you the starting value.
4For a pyramid, name the unknown cells and write one equation for each rule "sum of two below = the cell above"; or use the top formula.
5For a grid, express every cell in terms of the top-left number using +1 across and +7 down.
6Solve the resulting linear equation and check by rebuilding the figure.
7To design your own trick, plan the steps so the unknown cancels or the final number can be read off.

Think-of-a-Number Tricks, Number Pyramids and Calendar Grids in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

A calendar trick

Suppose a friend picks a 2 x 2 block of dates on a calendar page and says the sum of the four dates is 52.

Which four dates did she pick?

Try it first, then show the solutionSolution

Let the top-left date be $a$ . The dates are $a, a + 1, a + 7, a + 8$ and their sum is $4 a + 16$ .
$4 a + 16 = 52$ gives $a = 9$ .
The dates are 9, 10, 16 and 17; check $9 + 10 + 16 + 17 = 52$ .

Answer: 9, 10, 16, 17

Memory hook: Across is +1, down is +7.

Doubling and paying

Imagine a game in which your coins double each round and then you must pay 24 coins; you start with 21 coins.

How many coins do you have after each of three rounds?

Try it first, then show the solutionSolution

Round 1: $2 \times 21 - 24 = 18$ .
Round 2: $2 \times 18 - 24 = 12$ .
Round 3: $2 \times 12 - 24 = 0$ . (Check with $8 x - 7 c = 168 - 168 = 0$ .)

Answer: 18, 12, then 0

Memory hook: Doubling then paying a fixed amount eventually takes everything if the payment is too high.

Tricks to remember Think-of-a-Number Tricks, Number Pyramids and Calendar Grids

Short memory aids. Say each one aloud once, then use it on a question the same day.

Let x be the number
Always begin a trick problem with "let the number be x" and simplify after every step.
Use it when: Any think-of-a-number puzzle.
Binomial coefficients in pyramids
3-row top: 1, 2, 1. 4-row top: 1, 3, 3, 1. The coefficients are the number of paths from each bottom cell to the top.
Use it when: Finding the top without building.
Across 1, down 7
In a calendar, moving right adds 1 and moving down adds 7.
Use it when: Grid questions.
Constant in, constant out
If x cancels in the last step, the answer is the same whatever you started with.
Use it when: Designing tricks.

Think-of-a-Number Tricks, Number Pyramids and Calendar Grids: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: A trick that always gives the same value · 1 mark · MCQBoard level

Think of a number, double it, add 6, divide by 2, then subtract the number you started with. The result is always: (a) 2 (b) 3 (c) 6 (d) the starting number

Try it first, then show the solutionSolution

$x \to 2 x \to 2 x + 6 \to x + 3 \to x + 3 - x = 3$ .

Answer: (b) 3

Example 2: Designing a trick · 2 marks · Short answerBoard level

Design a think-of-a-number trick whose result is always 7, and prove it with algebra.

Try it first, then show the solutionSolution

Steps: think of a number $x$ ; double it ( $2 x$ ); add 14 ( $2 x + 14$ ); halve ( $x + 7$ ); subtract the original number. (1 mark)
$x + 7 - x = 7$ , so the result is always 7. (1 mark)

Answer: Double, add 14, halve, subtract the starting number: always 7.

Example 3: Decoding the date trick · 3 marks · Short answerHarder board level

In the date trick (multiply the month by 5, add 6, multiply by 4, add 9, multiply by 5, add the day) a friend reports 880. Find the date. Why can the answer 1000 not come from a real date?

Try it first, then show the solutionSolution

Subtract 165: $880 - 165 = 715$ . (1 mark)
$715 = 100 M + D$ , so $M = 7$ and $D = 15$ : the 15th of July. (1 mark)
For 1000: $1000 - 165 = 835$ would mean month 8 and day 35, but no month has 35 days. (1 mark)

Answer: 15 July; 1000 would give a 35th day of August, which does not exist.

Example 4: Number pyramids · 3 marks · Short answerHarder board level

A 3-row pyramid has bottom row 7, x, 13 and top 50. Find x and complete the pyramid. For a 4-row pyramid with bottom row 2, 5, 3, 6 find the top without building it, and then verify by building it.

Try it first, then show the solutionSolution

Top $= a + 2 b + c$ : $7 + 2 x + 13 = 50$ gives $2 x = 30$ , $x = 15$ . Middle row: $22, 28$ ; top $50$ . (1 mark)
4-row top $= a + 3 b + 3 c + d = 2 + 15 + 9 + 6 = 32$ . (1 mark)
Build: row 2: $7, 8, 9$ ; row 3: $15, 17$ ; top: $32$ . (1 mark)

Answer: x = 15; the 4-row top is 32.

Example 5: Calendar grids · 4 marks · Long answerHarder board level

In a calendar a 2 x 2 grid has sum 76. Find the four dates. Then find a 3 x 3 grid with sum 144 and show that the sum is 9 times the middle number.

Try it first, then show the solutionSolution

2 x 2: $4 a + 16 = 76$ , $a = 15$ ; dates 15, 16, 22, 23. (1 mark)
3 x 3 with top-left $a$ : the nine dates are $a, a + 1, a + 2, a + 7, a + 8, a + 9, a + 14, a + 15, a + 16$ . Sum $= 9 a + 72$ . (1 mark)
$9 a + 72 = 144$ gives $a = 8$ ; dates 8, 9, 10, 15, 16, 17, 22, 23, 24. (1 mark)
The middle number is $a + 8 = 16$ and $9 \times 16 = 144$ ; in general $9 a + 72 = 9 (a + 8)$ . (1 mark)

Answer: 15, 16, 22, 23; and the 3 x 3 grid starting at 8; sum = 9 x middle.

Where students lose marks in Think-of-a-Number Tricks, Number Pyramids and Calendar Grids

The slips examiners see most often, and what to write instead.

Mistake: Testing a trick with one number only
One example does not prove it works for every number.
Wrong working
"I tried 5 and got 2, so the trick always gives 2."
Right working
$\frac{2 x + 4}{2} - x = 2$ for every $x$ .
Do this: Use a letter-number.
Usually costs: 2 marks.
Mistake: Not simplifying after each step
Unsimplified brackets hide the cancellation.
Wrong working
"(2x + 4)/2 - x = 2x + 4/2 - x"
Right working
$\frac{2 x + 4}{2} = x + 2$ (divide every term), so the result is 2.
Do this: Divide every term by 2 and then subtract.
Usually costs: 1 mark.
Mistake: Using the wrong coefficients for the pyramid top
The middle numbers are counted twice (three times in a 4-row pyramid).
Wrong working
"4-row top = a + b + c + d."
Right working
Top $= a + 3 b + 3 c + d$ .
Do this: Build a small pyramid of letters to check.
Usually costs: 2 marks.
Mistake: Taking +1 down and +7 across in a calendar
Moving right adds 1; moving down adds 7.
Wrong working
"3 x 3 grid: across I add 7 and down I add 1, so the first row is a, a + 7, a + 14 and the first column is a, a + 1, a + 2." This swaps the steps, so the dates in the grid are wrong (for example, a + 7 is written next to a instead of a + 1).
Right working
Across is +1 and down is +7, so the rows are: a, a + 1, a + 2; a + 7, a + 8, a + 9; a + 14, a + 15, a + 16.
Do this: Draw the grid and write the entries.
Usually costs: 1 mark.

Exam tips

Write each trick step in a numbered list with the algebraic expression beside it.
Always finish with a check: rebuild the pyramid or grid with the values found.
For decoding tricks, state the digit-position argument (the day has two digits, so the last two digits are the day).
Use the formula for the top of a pyramid after you have shown where the coefficients come from.
In puzzles with payments, write the coins after each round as expressions.

Practice questions on Think-of-a-Number Tricks, Number Pyramids and Calendar Grids

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · Assertion-reasonBoard level
Assertion (A): In a 2 x 2 block of a calendar with top-left date a, the sum of the four dates is 4a + 16.
Reason (R): The four dates are a, a + 1, a + 7 and a + 8.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
Add the four expressions.
Answer
(a). $a + (a + 1) + (a + 7) + (a + 8) = 4 a + 16$ .
Q2 · 2 marks · Very short answerBoard level
A 3-row number pyramid has bottom row 4, 9, 6. Find the numbers in the middle row and at the top.
Hint
Add neighbouring pairs.
Answer
Middle row: $4 + 9 = 13$ and $9 + 6 = 15$ . Top: $13 + 15 = 28$ (also $a + 2 b + c = 4 + 18 + 6 = 28$ ).
Q3 · 3 marks · Short answerHarder board level
The bottom row of a 4-row number pyramid is 3, x, 2x, 5 and the top number is 80. Find x and build the pyramid.
Hint
Top = a + 3b + 3c + d.
Answer
Top $= 3 + 3 x + 3 (2 x) + 5 = 8 + 9 x = 80$ , so $9 x = 72$ , $x = 8$ . Bottom row: 3, 8, 16, 5. Row 2: 11, 24, 21. Row 3: 35, 45. Top: 80.
Q4 · 4 marks · Long answerHOTS / competency
Prove this birthday trick: multiply the day D by 2, add 5, multiply by 50, add the month M. Subtract 250 from the result to read off D and M. Decode the answer 2661. Why can no one get an answer 3112?
Hint
(2D + 5) x 50 = 100D + 250. The month is at most 12.
Answer
$(2 D + 5) \times 50 = 100 D + 250$ ; adding $M$ gives $100 D + 250 + M$ . Subtracting 250 leaves $100 D + M$ , and since $M \leq 12$ has at most two digits, the last two digits are the month and the digits before them are the day. For 2661: $2661 - 250 = 2411$ , so day 24 and month 11: 24 November. For 3112: $3112 - 250 = 2862$ would give day 28 and month 62, but there is no month 62.
Q5 · 4 marks · Case-basedHOTS / competency
Case study: in a game the genie doubles Karim's coins each time he walks round a tree, and Karim pays the genie c coins each round. Karim starts with 35 coins.
(i) Show that after 3 rounds he has 8x - 7c coins, where x is the number he starts with. (1 mark)
(ii) If c = 40, find his coins after each round. (1 mark)
(iii) For what values of c does Karim end up with more than he started with after 3 rounds? (1 mark)
(iv) What happens if c = 35? (1 mark)
Hint
After each round: new = 2 x old - c.
Answer
(i) Round 1: $2 x - c$ ; round 2: $2 (2 x - c) - c = 4 x - 3 c$ ; round 3: $2 (4 x - 3 c) - c = 8 x - 7 c$ . (ii) $2 \times 35 - 40 = 30$ ; $2 \times 30 - 40 = 20$ ; $2 \times 20 - 40 = 0$ . (iii) $8 x - 7 c > x$ gives $7 x > 7 c$ , i.e. $c < x = 35$ . (iv) If $c = 35$ , then each round gives $2 \times 35 - 35 = 35$ : the number of coins never changes.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.What is the sum of a calendar 2 x 2 block with top-left a?Show answer
4a + 16.
2.What is the top of a 3-row pyramid with bottom row a, b, c?Show answer
a + 2b + c.
3.What is the top of a 4-row pyramid with bottom row a, b, c, d?Show answer
a + 3b + 3c + d.
4.In the date trick, what is subtracted from the answer?Show answer
165.
5.Why does a trick give a fixed answer?Show answer
The letter-number cancels in the last step.
6.How many coins does the genie puzzle leave after 3 rounds?Show answer
8x - 7c.

Think-of-a-Number Tricks, Number Pyramids and Calendar Grids: frequently asked questions

Can the date trick work for any day?

Yes, because the day is at most 31, which has two digits; the trick is designed so the day is placed in the last two digits.

How do I design my own calendar trick?

Choose a grid shape, write each date in terms of the top-left a (adding 1 across and 7 down), find the sum as an expression and solve for a when told the sum.

Do number pyramids always have a unique solution?

If the bottom row is fully known, yes. With unknowns, you need enough equations to find them all.

What is the Virahanka-Fibonacci sequence?

The sequence 1, 2, 3, 5, 8, ... in which each number is the sum of the two before it.