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A balance puzzle or weighing puzzle is a logic puzzle about balancing items—often coins—to determine which one has different weight than the rest, by using balance scales a limited number of times. The solution to the most common puzzle variants is summarized in the following table: [1]
The number 12565, for instance, has digit sum 1+2+5+6+5 = 19, which, in turn, has digit sum 1+9=10, which, in its turn has digit sum 1+0=1, a single-digit number. The digital root of 12565 is therefore 1, and its computation has the effect of casting out (12565 - 1)/9 = 1396 lots of 9 from 12565.
The pattern also works with multiples of 10, by starting at 1 and simply adding 0, giving you 10, then just apply every number in the pattern to the "tens" unit as you would normally do as usual to the "ones" unit. For example, to recall all the multiples of 7: Look at the 7 in the first picture and follow the arrow.
The pictures on the right show how to calculate 345 × 12 using lattice multiplication. As a more complicated example, consider the picture below displaying the computation of 23,958,233 multiplied by 5,830 (multiplier); the result is 139,676,498,390. Notice 23,958,233 is along the top of the lattice and 5,830 is along the right side.
The predecessor of a natural number (excluding zero) is the previous natural number and is the result of subtracting one from that number. For example, the successor of zero is one, and the predecessor of eleven is ten ( 0 + 1 = 1 {\displaystyle 0+1=1} and 11 − 1 = 10 {\displaystyle 11-1=10} ).
“I got my calculator out and calculated it five times,” he said. The $50,000 prize is one of 50 offered in the game, which has a top reward of $5 million. Man couldn’t find his lottery ticket.
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