Tag Archives: kaprekar

Junction Combination Theorem

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Suppose that, a_{k_1} + s_d(a_{k_1}) = a_{k_2} + s_d(a_{k_2}) = \dots = a_{k_r} + s_d(a_{k_r}) = N for k_1, k_2, \ldots k_r > 1.s_d(n) denotes the sum of the digits of n. Thus all of a_{k_1}, a_{k_2}, \ldots , a_{k_r} yield N. That is, N has r “generators”. Numbers with more than one generator (i.e. r>1) are called “Junction Numbers” by Kaprekar.

Let N-1 and N+1 be two “Junction Numbers”, such that they have equal number of digits, say, d digits. If P_1, P_2, \ldots be the generators of N-1 and R_1, R_2, \ldots be the generators of N+1.
Then,

1(0)_{(1)_d} P_1 \quad, \quad 1(0)_{(1)_d} P_2 \quad, \quad \ldots

(9)_{(1)_d} R_1 \quad, \quad (9)_{(1)_d} R_2 \quad, \quad \ldots

give the generators of 1(0)_{(1)_d} N.

Let us consider an example to understand this theorem.

Consider two Junction Numbers 519 (generated by both 498 and 507) and 521 (generated by both 499 and 508).

Here N-1 = 519 and N+1 = 521

\therefore P_1 = 498 \quad, \quad P_2 = 507
\therefore R_1 =499 \quad, \quad R_2 = 508

d = 3 = Number of digits in N-1 or N+1 or N.

\therefore (1)_d = (1)_3 = 111
Then according to the theorem,
1(0)_{111} 498 \quad, \quad 1(0)_{111} 507

(9)_{111} 499 \quad, \quad (9)_{111} 508

give the four generators of 1(0)_{111} 520.

1(0)_{111} 520 is a number with 115 digits

Kaprekar named this theorem as Kaprekar’s Last Theorem in 1962, when he was seriously ill and feared that his death was nearing. He miraculously recovered and named it as Junction Combination Theorem.

Smallest “Junction Numbers”

  • Smallest “Junction Number” with two generators is \mathbf{101}
  • Smallest “Junction Number” with three generators is \mathbf{10000000000001} or \mathbf{1(0)_{12}1}
  • Smallest “Junction Number” with four generators is \mathbf{1000000000000000000000102} or \mathbf{1(0)_{21}102}

Happy Birthday Kaprekar

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Today is 110th birthday of great Indian mathematician D. R. Kaprekar

Dattaraya Ramchandra Kaprekar (17 January 1905 – 1986) was an Indian recreational mathematician who described several classes of natural numbers. For his entire career (1930–1962) he was a schoolteacher at Nasik in Maharashtra. He published extensively, writing about such topics as recurring decimals, magic squares, and integers with special properties. Kaprekar was once laughed at by most contemporary Indian mathmaticians for his so-called ‘trivial’ play with numbers. It required G. H. Hardy to recognize Ramanujan while Kaprekar’s recognition came through Martin Gardner (he wrote about Kaprekar in his “Mathematical Games” column in March 1975 issue of “Scientific American”)

Here I would discuss a mathemagical trick re-disvovered by Kaprekar called “Gap Filling Process” (though claimed to be present in vedic mathematics)

Gap filling process

This process is magical one and will make you Mathemagician

Let, (a)_n stand for a repeated n times (called Repunit a).

Then, we shall denote (a)_n ^m for m-th power of (a)_n

(9)_n ^m can be obtained by remembering the expansion for (9)^m and inserting in the gaps between digits of expansion of (9)^m with the numbers (9)_{n-1} and (0)_{n-1} alternately, beginning from left to right. No gap is counted after the unit digit.

Let’s see an example:
Find the value of (99999)\times (99999)\times (99999) = (99999)^3 = (9)_5 ^3.

Even my scientific calculator fails to calculate this exact value !

We know 9^3 = 729

Then the gaps are: ----7----2----9
Now fill the blanks alternately with (9)_{5-1} = (9)_4 and (0)_{5-1} = (0)_4.
We get: 9999\textbf{7}0000\textbf{2}9999\textbf{9}
Hence, (99999)\times (99999)\times (99999) = 999970000299999