Turn arithmetic into a small investigation. Look for what changes, what stays fixed, and why a shortcut works.
6 original activities1 interactive model3 quick questions
TRY IT IN YOUR BROWSER
Build a number with five switches
Each switch adds a different place value. Predict which switches make 19, then test your idea.
Your total000000
Can you make 19?
Tap a value to switch it on or off. An on switch adds its number to the total. Keyboard users can Tab between switches and press Space.
Five switches represent 0 to 31. No score, timer or account is needed.
Interactive result appears here when scripts are enabled.
Predict. Try. Understand.
Pause at each question before opening the explanation. Examples are invented for learning unless stated otherwise.
IDEA 01
One digit, a different job
Does the 4 mean the same thing in 42 and 24?
TRY / THINK
Write 42 as four bundles of ten and two singles. Now reverse the digits. Count the bundles again.
Explore the explanationWHY IT WORKS
A digit gets its value from its position. In base ten, moving one place left multiplies its value by ten. The symbol stays the same; its job changes. This is why a number is more than a bag of digits.
Worked example
42 = 4 × 10 + 2. Meanwhile 24 = 2 × 10 + 4. Their difference is 18, which is two groups of nine.
TAKE IT INTO PLANNING
A quantity of 24 cases is different from 24 individual items. Always check the unit as carefully as the number.
IDEA 02
A useful check that can still miss mistakes
If two totals have the same digit sum, must they be equal?
TRY / THINK
Add the digits of 247, then subtract that sum from 247. Repeat with 583. Finally compare 247 and 427.
Explore the explanationWHY IT WORKS
A whole number and its digit sum leave the same remainder when divided by nine. Each power of ten leaves remainder one. This makes digit sums useful for spotting some arithmetic errors, but rearranged digits can pass the check.
Worked example
247 − (2 + 4 + 7) = 234 = 26 × 9. Both 247 and 427 have digit sum 13, although the values are different.
TAKE IT INTO PLANNING
A quick validation rule is not a full reconciliation. Passing one check does not prove a forecast or an inventory record is correct.
IDEA 03
Make an awkward addition friendly
Which is easier: 68 + 32 + 27, or 100 + 27?
TRY / THINK
Choose a whole number from 1 to 99. Find what must be added to reach 100. Repeat with three choices.
Explore the explanationWHY IT WORKS
A complement is the amount missing from a chosen total. Pairing complements does not change the sum; it simply groups the work differently. The shortcut is supported by ordinary addition, not a mysterious prediction.
Worked example
For 68, the complement to 100 is 32. So 68 + 32 + 27 becomes 100 + 27 = 127.
TAKE IT INTO PLANNING
If a shipment needs 100 units and 68 are ready, the remaining requirement is 32. Check whether both figures use the same time window.
IDEA 04
Multiply by breaking a number apart
Can you calculate 23 × 14 using only doubling and addition?
TRY / THINK
Double 23 repeatedly to make 23, 46, 92 and 184. Select the entries that correspond to 14 = 8 + 4 + 2.
Explore the explanationWHY IT WORKS
Multiplication distributes over addition. Splitting 14 into powers of two turns one multiplication into a few easier ones. This is an established arithmetic method; the important step is keeping the matching weights.
Worked example
23 × 14 = 23 × (8 + 4 + 2) = 184 + 92 + 46 = 322.
TAKE IT INTO PLANNING
Break a large order into verified batches. The final total is reliable only if each selected batch is counted exactly once.
IDEA 05
Five yes-or-no answers, 32 possibilities
How can five switches describe every whole number from 0 to 31?
TRY / THINK
Use the switches above. Turn on 16, 2 and 1. Then try to make 30 with as few changes as possible.
Explore the explanationWHY IT WORKS
Each switch can be off or on, giving 2 × 2 × 2 × 2 × 2 = 32 combinations. Their values are 16, 8, 4, 2 and 1. Each combination gives a different total in this range.
Worked example
19 = 16 + 2 + 1, written 10011 in five-place binary. 30 = 16 + 8 + 4 + 2, written 11110.
TAKE IT INTO PLANNING
Digital systems can store yes/no conditions compactly. A code still needs a definition so people know what each position means.
IDEA 06
A change that preserves something
Can you make an odd count even by adding two repeatedly?
TRY / THINK
Start at 7 and add 2 four times. Repeat from 8. Label every result odd or even.
Explore the explanationWHY IT WORKS
Adding two preserves parity: odd stays odd and even stays even. Something that remains unchanged under an allowed move is called an invariant. Invariants can prove a proposed goal is impossible under those rules.
Worked example
7 → 9 → 11 → 13 → 15 remains odd. To reach 16, the allowed moves must change or include an odd adjustment.
TAKE IT INTO PLANNING
If stock can move only in pairs, a leftover single unit needs a different process. State the rules before trying to solve the puzzle.
A SMALL CHECK, NO PRESSURE
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The explanations, examples, questions and diagrams on this page were written for Supply Chain Ideas. The underlying principles are established science and mathematics, not claims of invention. The following references support further study; their text and illustrations are not reproduced.
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