Number Builder

What you see

Each trial is one chain. A chain has three to five numbers, each a single digit from 1 to 9, joined by operators. Some of the numbers are blanks. To the right sits the target.

• Filling a blank: choose a blank, then one of the digits 1 to 9 below the chain, or type 1 to 9. The next empty blank is then selected. Clear (Backspace) empties the selected blank, and with several blanks the Left and Right arrow keys move between them.
• Left to right, always. The chain is worked out strictly from left to right, with no operator precedence. The How it works notes say so, and it is the most common source of wrong answers.
• Status line: once every blank is filled, the line under the board works the chain step by step, for example "8 × 3 = 24, then 24 − 6 = 18, then 18 + 4 = 22", and the = before the target becomes ≠ if the chain does not make it. If a division does not come out exactly, the line says so ("14 ÷ 4 does not divide exactly"); if a step drops below zero where that is not allowed, it shows the step ("4 − 5 = −1 goes below zero"); and if a number repeats where every number must differ, it names it ("4 appears more than once").
• Rules on screen: a line above the chain names any extra rule. Levels 1 to 3 show "Rule: no step below zero", levels 4 to 8 show "Rules: all numbers different, no step below zero", and levels 9 and 10 show "Rule: all numbers different". Where every number must differ, digits already in the chain are dimmed below it.
• Division is always exact in the intended answer, and targets stay between −999 and 999.

A practice session opens at difficulty 3 and adapts as you go. There is no countdown, but a clock records your time. Press Enter to check.

How the levels change

The generator builds a valid chain first, then hides some digits, then checks every possible filling of the blanks. From level 3 it keeps the puzzle only if exactly one filling works. On levels 1 and 2 several fillings may work, and any of them is accepted. Operators are chosen at random from the level's set, so a level-6 chain may or may not contain a division.

What it measures

Number Builder feeds the keeping options open facet of your planning score, weighted 18% on the planning shelf, alongside Numerosity at 14%. In our game's own definition it is mostly numerical reasoning with a planning component, which is why its planning weight is lower than the pure planning games.

The planning demand comes from interacting constraints. Each blank's value depends on the others, the all-different rule links every blank to every fixed digit, and a single wrong early choice makes the rest unsolvable. This is a small constraint-satisfaction problem, and the classic advice from that field applies: decide the most constrained variable first, the fail-first principle.

It is a close relative of arithmetic target games such as the numbers round of the British quiz show Countdown, in which contestants use six numbers and the four basic operations to reach a random three-digit target in 30 seconds, with only exact division and no fractions or negative numbers allowed. Number Builder is narrower. The order of numbers and operators is fixed, so you only choose the missing digits, and the target is always exactly reachable.

Research on how people solve arithmetic shows that adults choose among strategies according to their speed and accuracy on the problem in front of them, and do better when they are allowed to choose than when they are not. The same flexibility pays here: some chains are fastest forwards, most are fastest backwards from the target.

How it is scored here

Scoring is all or nothing. The checker fills your digits into the chain and tests the rules directly: the chain must hit the target, every division must be exact, the all-different rule must hold where it applies, and no step may drop below zero on levels 1 to 8. A filling that passes is Correct and earns full credit. Anything else is Incorrect and earns nothing, and the line under the board then shows one correct chain, for example "One answer: 8 × 3 − 6 + 4 = 22".

In an adaptive session, Correct moves you up one level and Incorrect moves you down one. There is no "Partly correct" in this game.

Your game score adds a small time component (15% of the weight). Expected thinking time for this game starts at about 6 seconds and grows by about 2.4 seconds per level, so careful working on a hard chain costs little. On our scale, 50 means you are hitting targets at the rate your difficulty predicts. Your report shows targets reached exactly as the headline figure.

Where you will meet it

Arithmetic target puzzles appear in two forms in hiring and admissions. The first is ordinary quantitative aptitude, where working backwards from a required result is a standard time-saver. The second is interactive. Candidates report a "Digit Challenge" in Capgemini's game-based round, and Aon lists a digitChallenge in its smartPredict suite, which it describes as measuring numeracy. We have not found vendor documentation of a game identical to Number Builder, so treat it as training for the family of skills, not a replica of any test.

The keeping options open habit also transfers directly to arrangement and scheduling sets in CAT, bank exams and campus tests, where you fix the most constrained item first. The where planning is tested lesson has a worked example.

The method

1. Bracket the chain

   Rewrite the chain in your head with brackets from the left: blank × 3 − blank + 4 becomes ((blank × 3) − blank) + 4. This removes any temptation to multiply first.

2. Peel fixed steps off the end

   Start at the target and undo the last operation if its number is fixed. "+ 4 = 22" means the value before it was 18. Undo means: subtract for +, add for −, divide for × (it must divide exactly), multiply for ÷. Keep going until you reach a blank.

3. Turn the first blank you meet into a range

   When the step involves a blank, write the range it allows. "z − c = 34" with c from 1 to 9 means z is 35 to 43.

4. Cut the range with the other side

   Now look at what the fixed part of the chain can produce going forwards. If z comes from "something × 7", z must be a multiple of 7, so only 35 or 42 survive. Two constraints meeting usually leave one or two cases.

5. Try the cases, most constrained blank first

   For each surviving case, fill the blank that has fewest options, usually the one next to a × or ÷. Then check the all-different rule against every number in the chain, fixed digits included.

6. Confirm with the status line, then submit

   Enter your digits and read the steps on the line under the board. If the last step makes the target and the sign reads =, you are done. If the line says a division "does not divide exactly", a step "goes below zero" or a number "appears more than once", you broke a rule, not just the arithmetic.

Worked example 1: two blanks and the all-different rule

Show the solution

Bracket it: ((a × 3) − b) + 4 = 22.

Peel the fixed step: undo + 4, so (a × 3) − b = 18.

Range: b is 1 to 9, so a × 3 is 19 to 27, and it must be a multiple of 3: 21, 24 or 27. That gives three pairs: a = 7, b = 3; a = 8, b = 6; a = 9, b = 9.

Apply the rules: 7 and 3 repeats the fixed 3. 9 and 9 repeats itself. Only 8 and 6 keeps all four numbers different.

This is keeping options open in miniature. If you had started by trying a = 7 because 7 × 3 = 21 "looks close", you would find b = 3, see the chain make 22, and still be marked wrong.

Worked example 2: three blanks, worked backwards

Show the solution

Bracket it: (((3 + a) ÷ b) × 7) − c = 34. Call the value before the last step z, and the value before × 7 y.

1. Last step: z − c = 34 with c from 1 to 9, so z is 35 to 43.
2. Cut with × 7: z = y × 7, so z is 35 or 42, which makes y equal to 5 (with c = 1) or 6 (with c = 8).
3. Try y = 5: (3 + a) ÷ b = 5, and 3 + a is between 4 and 12, so it is 5 or 10. If 5, then a = 2 and b = 1, but c is already 1. If 10, then a = 7, but 7 is a fixed digit. Both fail.
4. Try y = 6: 3 + a is 6 or 12. If 6, then a = 3, a repeat. If 12, then a = 9 and b = 2. Check: 3, 9, 2, 7, 8 are all different.

The blanks are 9, 2, 8. Notice that you never guessed a digit. Every choice came from narrowing a range, which is why working backwards is faster than trying digits from the left.

Tips and traps

Levelling up

The chain grows from three numbers to five, blanks from one to three, and the operator set from addition alone to all four operations. From level 4, all numbers must differ, which is where the planning starts to bite. At level 9 the "no step below zero" rule is lifted, and at level 10 addition disappears entirely.

• Levels 1 to 3: one blank. Solve it by undoing the chain from the target; no guessing needed.
• Levels 4 to 6: watch the all-different rule. When two blanks interact, turn the later one into a range first.
• Levels 7 and 8: three blanks and five numbers. Peel off every fixed step at the end, then look for the division.
• Levels 9 and 10: intermediate values can be negative, so a subtraction early in the chain no longer rules anything out. Keep the signs straight: a negative value times a positive digit stays negative.

1. Week 1
   • Three sessions of 10 minutes
   • Bracket every chain from the left before touching the digits
   • Target: no answers lost to precedence mistakes
2. Week 2
   • Three sessions
   • Solve every chain backwards from the target first
   • Target: check all-different against fixed digits every time
3. Week 3
   • Two Number Builder and two Numerosity sessions
   • Start from the most constrained blank, usually next to × or ÷
   • Note which rule caused each miss
4. Week 4
   • Four shorter sessions on consecutive days
   • At levels 7 and above, write the range for each blank before filling any
   • Compare your targets-reached rate with week 1

Check yourself

5 practice questions

1. In Number Builder, what does the chain 4 + 2 × 3 make?
2. At level 5 the chain is blank × 3 − blank + 4 = 22. Why is 7 and 3 wrong, even though it reaches 22?
3. At level 10 the chain is 6 ÷ blank − blank − 2 × blank = −27. Which filling works?

and 2 more

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Sources

1. Haralick, R. M., & Elliott, G. L. (1980). Increasing tree search efficiency for constraint satisfaction problems. Artificial Intelligence, 14(3), 263–313. doi:10.1016/0004-3702(80)90051-X
2. Countdown (game show): numbers round rules. Wikipedia
3. Siegler, R. S., & Lemaire, P. (1997). Older and younger adults' strategy choices in multiplication: Testing predictions of ASCM using the choice/no-choice method. Journal of Experimental Psychology: General, 126(1), 71–92. doi:10.1037/0096-3445.126.1.71
4. PrepInsta (prep site, candidate-reported). Capgemini game-based aptitude. prepinsta.com
5. cut-e (now Aon). smartPredict product page, archived 10 December 2019. Wayback Machine

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