New Scientist Enigma 527 – Sum Secret

by Barry Clarke

From Issue #1679, 26th August 1989

Oliver Oddwelly ws a cringing wreck of a man. He frequently needed to have his safe combination number, his bank account number, and his credit card number on hand, but was too frightened to write them down anywhere in case someone found them.

However, one day, just as he was oiling the padlock on the fridge, he suddenly hit upon an ingenious way of concealing the numbers. He decided to compose an arithmetical problem, the solution to which would reveal three seven digit numbers he needed to remember.

The sum he invented is shown below, where the first row added to the second gives the third, the fourth subtracted from from the third gives the fifth, and the fifth added to the sixth gives the seventh. One digit can be erased from each row (not necessarily the same position in each row) and the gaps can be closed up to leave three columns of digits, then a second digit can be rubbed out in the same way to give two columns, then a third to leave one column, so that a valid sum remains each time. The three sets of seven digits erased (read down the columns) respectively reveal the numbers he had to remember. What were the three numbers ?

\[\begin{array}{r} 6897\\ +2968\\ \hline 9865\\ – 4968\\ \hline 4897\\ + 3856 \\ \hline 8753\\ \hline \end{array}\]