Henry Ernest Dudeney/Modern Puzzles/60 - Digital Coincidences/Solution

Modern Puzzles by Henry Ernest Dudeney: $60$

Digital Coincidences

If I multiply, and also add, $9$ and $9$, I get $81$ and $18$, which contain the same figures.

If I multiply and add $2$ and $47$ I get $94$ and $49$ -- the same figures.

If I multiply and add $3$ and $24$ I get the same figures -- $72$ and $27$.

Can you find two numbers that, when multiplied and added will, in this simple manner, produce the same three figures?

Solution

Dudeney offers up:

\(\ds 2 + 497\)

\(=\)

\(\ds 499\)

\(\ds 2 \times 497\)

\(=\)

\(\ds 994\)

and:

\(\ds 2 + 263\)

\(=\)

\(\ds 265\)

\(\ds 2 \times 263\)

\(=\)

\(\ds 526\)

Also see

• Numbers which Multiplied by 2 are the Reverse of when Added to 2

Historical Note

Martin Gardner notes that Harry Lindgren points out the fact that by inserting $9$s you can obtain answers with any number of digits:

\(\ds 2 + 4997\)

\(=\)

\(\ds 4999\)

\(\ds 2 \times 4997\)

\(=\)

\(\ds 9994\)

\(\ds 2 + 2963\)

\(=\)

\(\ds 2965\)

\(\ds 2 \times 2963\)

\(=\)

\(\ds 5926\)

\(\ds 2 + 49997\)

\(=\)

\(\ds 49999\)

\(\ds 2 \times 49997\)

\(=\)

\(\ds 99994\)

\(\ds 2 + 29963\)

\(=\)

\(\ds 29965\)

\(\ds 2 \times 29963\)

\(=\)

\(\ds 59926\)

and so on.

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $60$. -- Digital Coincidences
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: $111$. Digital Coincidences